THE  KINETIC  THEORY 


APPLICATIONS  OF 

THE- KINETIC  THEORY 

TO   GASES,  VAPORS,  PURE   LIQUIDS,  AND 
THE   THEORY    OF   SOLUTIONS 


BY 


WILLIAM    PINGRY   BOYNTON,    PH.D.      ^ 

Assistant  Professor  of  Physics  in  the  University  of  Oregon 


Yotfc: 
THE   MACMILLAN   COMPANY 

LONDON:  MACMILLAN  &  CO.,  LTD. 
1904 


QC/7S 
37 


Copyright,  1904 
BY  THE  MACMILLAN  COMPANY 


Set  up,  electrotyped  and  printed  March,  1904 


PRESS  or 

THE  NEW  ERA  PRINTING  COMMIT, 
LANCASTER.  PA. 


PREFACE. 

This  book  presupposes  a  moderate  acquaintance  with 
the  fundamentals  of  physics  and  chemistry,  and  a 
mathematical  equipment  involving  familiarity  with  the 
differential  calculus  and  at  least  the  notation  of  the 
integral  calculus.  It  embodies  a  course  of  lectures 
given  at  the  University  of  California  during  the  aca- 
demic years  1898-1901,  but  includes  for  the  sake  of 
greater  completeness  of  treatment  some  matter  not 
given  in  the  lectures.  For  detailed  information  regard- 
ing many  of  the  topics  mentioned,  as  for  instance 
osmostic  pressure,  and  electrolysis,  and  for  a  state- 
ment of  the  experimental  basis  for  the  theories  pre- 
sented the  reader  is  referred  to  the  standard  texts  which 
treat  of  these  topics  in  extenso.  The  intent  in  this 
volume  is  not  so  much  to  discuss  these  facts  and 
theories  by  themselves  as  to  present  their  possible  or 
probable  relations  to  each  other  in  the  light  of  the 
kinetic  theory. 

While  any  adequate  treatment  of  the  kinetic  theory 
must  be  mathematical,  and  the  authoritative  treatises 
put  forth  by  the  creators  and  masters  of  the  theory 
make  severe  demands  upon  the  attainments  of  him  who 
would  read  them,  the  theory  itself  owes  its  interest  and 
value  to  the  fact  that  it  is  fundamentally  a  physical  and 
not  merely  a  mathematical  presentation,  that  it  is  not 
satisfied  with  coordinating  external  phenomena  by  the 
formulation  of  geometrical  or  algebraic  laws,  but  at- 


vi  PREFACE. 

tempts  to  present  to  the  imagination  the  mechanism  by 
which  things  take  place.  The  fact  that  the  behavior 
of  gases,  or  the  laws  of  dilute  solutions,  or  of  electrol- 
ysis can  be  presented  by  a  system  of  equations  which 
make  no  mention  of  molecules,  atoms  or  ions,  is  no 
argument  for  or  against  their  existence.  Such  a  sys- 
tem affords  a  beautiful  example  of  a  mathematical 
theory,  but  can  never  fill  the  place  of  a  physical  theory. 

Because  it  is  a  physical  theory,  the  kinetic  theory 
must  face  not  only  the  problems  of  the  gaseous  state, 
but  also  of  the  liquid  and  solid  states  ;  not  only  the 
problem  of  pure  substances,  but  also  those  of  mixtures 
and  solutions.  To  say  that  it  has  mastered  all  these 
problems  is  manifestly  absurd ;  yet  it  seems  desirable 
to  present  a  treatment  of  as  large  a  part  of  the  field  as 
possible  for  the  sake  of  symmetry  and  perspective. 

The  author  entirely  disclaims  any  originality  either 
in  material  or  in  the  treatment  of  the  subjects  con- 
sidered. His  part  has  been  to  collect  and  to  attempt 
a  systematic  presentation.  He  has  attempted  to  give 
credit  to  the  sources  of  his  information,  referring  to  the 
original  papers  wherever  he  has  Had  access  to  them, 
or  could  learn  of  the  original  source. 

It  is  a  pleasure  to  acknowledge  here  my  indebted- 
ness to  the  lectures  of  Prof.  A.  G.  Webster,  and  to  the 
section  on  the  kinetic  theory  by  Jaeger,  in  Winkel- 
mann's  "  Handbuch  der  Physik."  My  wife  has  given 
invaluable  assistance  by  copying  all  the  manuscript. 

EUGENE,  OREGON, 
January  19,  1904. 


CONTENTS. 
CHAPTER   I. 


INTRODUCTION 


CHAPTER   II. 

IDEAL  GASES... 7 

Statement  of  Assumptions,  7.  Law  of  Pressure,  9. 
Computation  of  Velocities,  14.  Boyle's  Law  and 
Temperature  Scale,  14.  Velocity  Lines,  17.  Max- 
well's Velocity  Law,  21.  Meaning  of  a,  26.  Average 
Speed,  28.  "  Mean  Square  "  Speed,  29.  Discussion 
of  Law  of  Pressure,  31.  Mixtures  of  Gases,  38.  Dai- 
ton's  Law,  39.  Temperature,  42.  Avogadro's  Law, 
43.  Thermodynamics,  43.  First  Law,  43.  Specific 
Heats,  44.  Adiabatics,  47.  Entropy,  51.  Second 
Law,  53.  Demon  Engine,  54. 

CHAPTER    III. 

GASES  WHOSE  MOLECULES  HAVE  DIMENSIONS 55 

Mean  Free  Path,  55.  Relative  Speed,  60.  Num- 
ber of  Molecules  Travelling  a  Given  Distance,  64. 
Effect  on  Pressure,  67.  Ratio  of  Specific  Heats,  71. 
Boltzmann's  Theorem,  75. 

CHAPTER   IV. 
TRANSPORT  PROBLEMS 79 

Conduction  of  Electricity,  79.     Viscosity  of  Gases, 

85.     Coefficient  of  Viscosity,  89.     Dependence  upon 

Temperature  and  Pressure,  91.     Conduction  of  Heat, 

92.     Correction  for  Small  Pressures,  95.     Diffusion, 

vii 


viii  CONTENTS. 

96.  Diffusion  into  Itself,  98.  Collisions  in  Mixed 
Gases,  100.  Free  Path  in  Mixed  Gases,  103.  Co- 
efficient of  Diffusion,  1 03.  Simplified  Coefficient,  1 05 . 

CHAPTER  V. 

CHANGE  OF  STATE 108 

General  Phenomena,  108.  Water  and  Steam 
Lines,  no.  Critical  Point,  no.  Model,  in.  Ther- 
modynamics, 112.  Thomson' si  deal  Isothermal,  117. 

CHAPTER   VI. 

EQUATION  OF  VAN  DER  WAALS 120 

Restrictions  removed,  120.  Equation  of  van  der 
Waals,  122.  Other  Equations,  123.  Pressure  and 
Volume  Coefficients,  126.  Thermodynamics,  .128. 
Formulae,  132.  Ratio  of  Specific  Heats,  134.  Form 
of  Isothermals,  135.  Critical  Point,  138.  Corre- 
sponding States,  140.  Critical  Data,  143.  Discussion 
of  Critical  Volume,  145.  Dieterici's  Equation,  147. 
Berthelot's  Modification,  150. 

CHAPTER  VII. 
VAPORIZATION 152 

Traube's  Method,  152.  Dieterici's  Method,  154. 
Capable  Molecules,  156.  Number  Passing  from 
Liquid  to  Vapor,  157.  Energy  Carried  by  Them,  1 58. 
Momenta,  160.  Relations  between  Speeds  and  Num- 
bers of  Molecules  in  Liquid  and, Vapor,  163.  Tem- 
perature and  Speeds,  164.  Review  of  Assumptions, 
164.  Ratio  of  Covolumes,  165.  Latent  Heat,  167. 
Dieterici's  Equation,  171. 

CHAPTER  VIII. 
MOLECULES  WITHIN  A  LIQUID !74 

Failure  of  Gas  Laws,  174.  Mean  Free  Path,  175. 
Value  of  d,  179.  Space  Occupied  by  Molecules,  179. 


CONTENTS.  ix 

Formula  for  Pressure,  180.  Covolume,  181.  Inter- 
nal Pressure  Related  to  Surface  Tension  and  Coeffi- 
cient of  Compressibility,  182. 

CHAPTER   IX. 
SOLUTIONS 185 

Mixtures  of  Gases,  185.  Absorption  of  Gases,  189. 
Henry's  Law,  189.  Effect  of  Temperature,  191. 
Solution  of  Liquids,  192.  Vapor  Over  Mixed  Liquids, 
193.  Distillation,  198.  Osmosis,  199.  Osmotic  Pres- 
sure, 200.  Follows  Gas  Laws,  202.  Relation  to 
Vapor  Pressure,  206.  Boiling  Point,  208.  Freezing 
Point,  209.  Raoult's  Law,  212.  Thermodynamics, 

213- 

CHAPTER   X. 

KINETIC  THEORY  OF  SOLUTIONS 216 

Surface  Film,  216.  Form  of  Equation,  217.  Par- 
tial Pressures,  218.  Latent  Heat,  219.-  Heat  of 
Dilution,  222.  Osmotic  Pressure,  222. 

CHAPTER   XI. 

DISSOCIATION  AND  CONDENSATION 224 

Double  Decomposition,  224.  Dissociation,  224. 
Boltzmann's  Theory  of,  225.  Alternative  Theory, 
226.  Effect  of  Density,  231.  Resulting  Density, 
233.  Experimental  Verification,  236.  Polymeriza- 
tion of  Water,  237.  Electrolytic  Dissociation,  238. 
Ions,  239.  Faraday's  Laws,  242.  Explanation  of 
Electrolysis,  243.  Molecular  Conductivity,  244. 
Speed  of  Ions,  245.  Dissociation  Constant,  247. 
Effect  of  Water,  249.  The  Galvanic  Cell,  250. 
Solution  Pressure,  252.  Influence  of  Solvent,  256. 
Dissociation  of  Water,  257.  lonization  of  Gases,  257. 
Corpuscles,  259.  Condensation  Nuclei,  261.  Elec- 
tric Spark,  262.  Production -and  Removal  of  Cor- 
puscles, 263. 


X  CONTENTS. 

CHAPTER   XII. 

SUMMARY 265 

Maxwell's  Distribution,  265.  H  Theorem,  266. 
Entropy  and  Second  Law,  268.  Degrees  of  Freedom, 
268.  Escape  of  Gases  from  Atmosphere,  270.  Law 
of  Force  between  Molecules,  271.  Mean  Free  Path, 
274.  Dimensions  of  Molecules,  276.  Thin  Films,  277. 
Number  of  Molecules,  279.  Ionic  Charge,  280. 
INDEX...  ...281 


KINETIC  THEORY. 


CHAPTER    I. 
INTRODUCTION. 

IN  the  so-called  "  Kinetic  Theory"  an  attempt  is 
made  to  explain  the  inner  and  invisible  relations  of 
matter  in  a  way  which  shall  satisfactorily  account  for 
and  describe  the  phenomena  which  are  actually  ob- 
served. 

From  the  earliest  time  it  has  been  recognized  that 
there  was  some  relation  between  heat  and  motion. 
The  primitive  method  of  obtaining  fire  by  friction 
between  two  pieces  of  wood  is  evidence  of  this,  and 
references  to  the  works  of  writers  of  the  Middle  Ages 
can  be  given  which  show  the  same  general  idea.  The 
first  expression  of  a  fairly  clear  conception  of  the  ideas 
which  lie  at  the  basis  of  the  Modern  Kinetic  Theory 
is  probably  to  be  found  in  Daniel  Bernouilli's  "Hydro- 
dynamica"  which  appeared  in  1738. 

The  real  foundations  of  the  modern  mathematical 
form  of  the  Kinetic  Theory  were  laid  by  Joule  in 
1848,  and  by  Clausius,  Maxwell,  Boltzmann  and 
others.  At  first  the  attempt  was  made  to  explain  the 
properties  of  gases  only,  a  problem  which  seemed  the 


2  KINETIC   THEORY. 

more  hopeful  because  of  the  simple  laws  which  relate 
the  phenomena  of  gases.  Of  late  years  the  treat- 
ment has  been  extended  with  some  success  to  liquids 
also.  We  shall  attempt  to  give  an  elementary  treat- 
ment of  the  Kinetic  Theory  both  of  liquids  and  gases. 

The  object  of  our  treatment  is  not  argumentative, 
so  that  for  the  present  it  is  immaterial  whether  the 
theory  be  regarded  as  a  statement  of  what  actually 
occurs,  or  simply  as  a  mechanical  analogy,  a  model, 
if  you  please,  to  help  us  understand  the  external 
phenomena.  Yet  if  the  analogy  be  close  enough  and 
extend  far  enough  we  hold  ourselves  free  to  accept  it 
as  more  than  a  simple  analogy. 

The  Kinetic  Theory  may  be  regarded  as  a  Special 
or  Specialized  Theory  of  Heat,  while  Thermodynamics 
is  a  more  general  theory,  assuming  only  that  heat  is 
a  form  of  energy,  but  not  describing  further  the  par- 
ticular form.  Hence  all  the  theorems  of  Thermody- 
namics must  be  consistent  with  the  results  which  we 
shall  obtain,  and  some  of  them  may  appear  to  derive 
their  validity  from  causes  which  we  shall  unravel. 

Certain  general  notions  which  belong  to  the  Kinetic 
Theoiy  may  be  stated  at  the  outset.  Divisibility  is 
ordinarily  stated  to  be  one  of  the  properties  of  matter. 
Divisibility  to  an  indefinite  extent,  or  infinite  divisibility, 
to  use  the  shorter  term,  is  a  purely  mathematical  con- 
ception. Anything  which  is  continuous,  as  space,  or 
time,  can  be  thought  of  as  infinitely  divisible,  or 
divisible  at  any  point  indifferently.  Physicists  and 
chemists  have  generally  agreed  to  consider  that  a  sub- 


INTRODUCTION.  3 

stance  may  be  divided  into  very  small  parts  without 
losing  its  identity  as  a  substance.  The  smallest  parts 
which  can  still  retain  this  identity  are  called  Molecules. 
These  molecules  may  be  yet  further  divisible  into 
Atoms,  which,  however,  are  different  in  their  properties 
from  the  molecules  which  they  compose.  The  atoms 
themselves  are  regarded  as  indivisible.  All  the  mole- 
cules of  a  pure  substance  are  regarded  as  being  just 
alike  in  all  respects,  though  the  molecules  of  different 
substances  are  different.  Similarly  all  the  atoms  of 
one  kind  are  alike  in  all  respects,  though  there  are 
many  kinds  of  atoms.  This  theory  has  been  pro- 
pounded to  explain  the  facts  of  chemistry,  and  particu- 
larly the  fact  of  combination  in  definite  proportions. 
We  shall  find  it  however  a  convenient  starting  point 
for  our  work. 

We  shall  think  then  of  a  gas  as  composed  of  a 
great  number  of  particles,  or  molecules,  these  molecules 
being  for  any  one  gas  all  just  alike,  though  we  may 
find  it  convenient  sometimes  to  consider  mixtures,  in 
which  there  will  be  several  sets  of  molecules  of  differ- 
ent kinds,  but  all  the  molecules  of  any  one  kind  will 
be  just  alike.  These  molecules  will  be  subject  to  the 
laws  of  mechanics  ;  that  is,  Newton's  laws  and  their 
consequences  can  be  applied  to  them  just  as  to  ordi- 
nary objects. 

If  we  try  to  picture  to  ourselves  the  behavior  and 
motions  of  these  molecules,  we  have  to  imagine  them 
magnified  to  a  very  great  extent,  and  then  to  con- 
sider what  takes  place  in  a  space  which  is  really  very 


4  KINETIC   THEORY. 

minute.  Such  a  space  is  represented  by  Fig.  i.  We 
may  think  of  this,  if  we  choose,  as  representing  the 
positions  of  the  molecules  of  a  very  small  portion  of 
a  solid.  These  molecules  are  to  be  thought  of  as  all 

&  a  in  violent  motion,  but  they  are 

subject  to  mutual  attractions, 
g  9  and  possibly  to  repulsions  also. 

e  It  may  be  that  in  a  solid  the 

•  0  molecules  occupy  a  very  much 

e  larger  part  of  the  space  than 

•  is  here  represented.  It  is  very 

probable  that  they  are  not  sim- 
ple in  form.  But  we  can  think  of  the  points  as  repre- 
senting the  mean  positions  of  the  centers  of  the  mole- 
cules. Suppose  then  that  some  molecule,  as  a,  starts 
to  move  to  the  right.  It  will  be  opposed  in  that  motion 
by  the  attractions  of  b  and  r,  and  by  d  and  e,  which  it 
may  hit,  and  rebound,  or  they  may  simply  exert  a  re- 
pulsive force  when  a  gets  too  near  them.  Whatever 
may  be  the  causes,  however,  a  soon  starts  back,  swings 
perhaps  past  its  first  position,  only  to  be  sent  back 
and  to  oscillate  in  some  path  simple  or  complex,  never 
getting  far  from  its  original  position. 

If,  however,  a,  going  to  the  right  manages  to 
escape  between  d  and  e,  it  may  not  come  back  at  all 
to  its  first  place,  but  may  wander  now  to  one  part, 
now  to  another,  of  the  body.  If  a  great  many,  or  all 
the  molecules  have  this  freedom  of  motion,  we  have 
a  picture  of  the  liquid  state.  In  the  solid  state  very 
little  diffusion  can  take  place,  but  of  liquids  diffusion 


INTRODUCTION.  5 

is  an  especial  characteristic,  as  much  so  as  their 
mobility. 

In  the  liquid  state  as  we  have  pictured  it,  there  is 
still  an  attractive  force  between  the  molecules,  that  is, 
liquids  exhibit  cohesion,  and  the  molecules  never  get 
very  far  apart.  In  the  interior  of  a  liquid  this  cohe- 
sion exhibits  no  marked  effect,  except  as  in  connec- 
tion with  the  other  properties  of  the  molecule  it  helps 
determine  the  density.  But  near  the  surface  a 
molecule  feels  the  attraction  of  the  molecules  on  one 
side  of  it,  while  there  are  fewer  molecules  on  the 
other,  so  that  there  is  an  unbalanced  tension  tending 
to  draw  it  toward  the  body  of  the  liquid,  or  at  least 
to  keep  it  from  escaping  from  the  liquid.  This  un- 
balanced tension  explains  why  a  liquid  can  have  a 
free  surface,  just  as  a  solid  can,  and  is  called  surface 
tension. 

Most  of  the  molecules  of  a  liquid  do  not  have  a 
sufficiently  high  velocity  when  they  approach  this 
surface  region  to  enable  them  to  break  through  and 
escape  from  the  liquid  in  spite  of  the  unbalanced  at- 
traction, or  surface  tension  ;  but  we  shall  see  later 
that  the  molecules  do  not  all  have  the  same  velocity, 
and  so  some  of  them  which  happen  to  have  velocities 
very  much  higher  than  the  average  may  escape 
through  the  surface  of  the  liquid  into  the  space  above. 
These  molecules  will  then  constitute  the  vapor  of  the 
liquid.  If  the  space  above  the  liquid  is  confined,  after 
a  time  some  of  the  free  molecules  coming  back  near 
the  surface  of  the  liquid  may  plunge  back  into  it; 


6  KINETIC   THEORY. 

this  is  condensation,  and  when  the  rate  at  which  the 
molecules  are  leaving,  and  the  rate  at  which  they 
return  to  the  liquid,  or  the  rates  of  evaporation  and 
condensation,  are  equal,  the  space  above  the  liquid  is 
said  to  be  saturated  with  the  vapor. 


CHAPTER   II. 
IDEAL   GASES. 

THE  volume  occupied  by  a  substance  in  the  gaseous 
state  is  in  general  so  very  much  greater  than  that 
which  it  occupies  in  the  solid  or  liquid  state  that  we 
are  led  to  believe  that  the  molecules  of  the  gas  them- 
selves occupy  a  very  small  portion  of  the  space  filled 
by  the  gas,  but  that  it  is  by  the  violent  motion  of 
these  molecules  that  the  gas  can  seem  to  fill  all  the 
space. 

If  we  can  neglect  or  disregard  the  space  occupied 
by  the  substance  of  the  molecules,  we  can  obtain 
some  quite  simple  relations.  For  the  sake  of  sim- 
plicity we  shall  also  disregard  the  effect  of  gravita- 
tion, though  we  may  later  ask  what  its  effect  may  be, 
and  we  shall  also  for  the  present  neglect  the  effect  of 
the  mutual  attraction  between  molecules  which  in 
liquids  and  solids  gives  rise  to  cohesion.  This  we  do 
on  the  assumption  that  it  is  only  for  a  very  small 
portion  of  the  time  that  a  molecule  is  near  enough  to 
any  other  molecule  to  be  perceptibly  affected  by  its 
attraction. 

Stated  more  formally,  we  shall  assume  that  the 
total  volume  of  the  molecules  themselves  is  so  small 
in  comparison  with  the  space  in  which  they  move  that 
it  can  be  entirely  disregarded  ;  that  the  time  during 

7 


8  KINETIC   THEORY. 

which  two  molecules  are  in  contact  with  each  other  is 
very  small  as  compared  with  the  average  time  during 
which  a  molecule  is  moving  between  successive  im- 
pacts, so  that  in  comparison  it  can  be  entirely  neg- 
lected, and  so  that  further  there  is  no  probability  of 
the  molecules  hitting  each  other  in  groups  of  more 
than  two,  that  is,  there  will  be  no  collisions  of  more 
than  two  molecules  together ;  and  that  there  are  no 
forces  acting  upon  the  molecules  except  those  that 
arise  from  and  act  during  collisions. 

The  molecules  will  move  in  straight  lines  with  uni- 
form speed,  except  when  their  direction  and  speed  are 
being  changed  by  a  collision.  The  speeds  of  different 
molecules,  and  the  speeds  of  the  same  molecule 
just  before  and  just  after  a  collision  may  be  very 
different. 

For  the  sake  of  definiteness  in  our  conceptions  and 
simplicity  in  our  reasoning  we  shall  for  the  present 
regard  the  molecules  as  smooth,  hard,  perfectly  elastic 
spheres.  Under  these  assumptions,  if  we  could  at  any 
time  know  exactly  the  positions  and  velocities  of  all 
the  molecules  of  a  gas,  it  would  be  only  a  matter  of 
comparatively  simple  computations  to  follow  the  path 
of  each  molecule,  to  determine  its  collisions  and  the 
velocities  resulting  from  these  collisions.  But  the 
vast  number  of  the  molecules  and  the  frequency  of 
their  collisions  makes  this  method  of  treatment  a  task 
of  hopeless  magnitude. 

The  other  possible  method  of  studying  these  mo- 
tions is  to  confine  our  attention  to  some  small  space, 


IDEAL   GASES.  9 

and  study  its  conditions  ;  some  molecules  leave  this 
space,  others  come  in  to  take  their  places ;  individual 
molecules  change  their  directions  and  speeds,  but 
others  take  the  directions  and  speeds  these  had.  Our 
study  then  becomes  a  matter  of  statistics,  of  numbers 
and  averages. 

We  may  however  very  simply  find  an  answer  to  the 
question,  what  pressure  will  be  exerted  by  a  given 
body  of  gas.1  We  shall  assume  that  the  body  of  gas 
we  are  studying  is  confined  within  a  closed  receptacle, 
whose  walls  may  be  considered  perfectly  smooth  and 
hard.  Imagine  a  small  portion  of  one  of  the  walls 
small  enough  so  that  it  may  be  regarded  as  plane, 
separated  from  the  rest  of  the  wall  so  as  to  be  capable 
of  a  slight  backward  and  forward  motion,  as  a  piston. 
The  continual  impact  of  the  molecules  of  the  gas  will 
tend  to  force  this  piston  back  away  from  the  space 
occupied  by  the  gas,  and  we  shall  try  to  find  what  force 
applied  to  the  back  of  the  piston  will  just  suffice  to 
hold  it  in  equilibrium  against  the  impact  of  the 
molecules. 

On  the  piston  as  a  base  erect  an  imaginary  cylinder 
of  height  //,  with  its  walls  perpendicular  to  the  piston, 
and  with  its  opposite  base  parallel  and  equal  to  it. 
The  phenomena  inside  this  cylinder  will  be  exactly 
the  same  whether  the  walls  of  the  cylinder  are  solid, 
so  that  the  same  molecules  rebound  and  come  back 
into  the  space,  or  whether  as  some  go  out  others 
come  back  in  to  occupy  the  same  space  with  the  same 

1  Joule,  Phil.  Mag.  (4),  14,  p.  21 1,  1857. 


10  KINETIC   THEORY. 

variety  of  velocities,  just  as  if  they  had  come  from  a 
region  which  was  the  exact  mirrored  image  of  the  region 
just  within  the  wall.  For  convenience  in  computation 
we  shall  regard  the  cylinder  as  actually  existing,  with 
smooth,  hard  walls,  restricting  the  motions  of  the 
molecules. 

For  convenience  we  shall  also  make  the  two  follow- 
ing assumptions,  which  we  know  are  not  in  accord 
with  the  facts,  that  all  the  molecules  are  moving  with 
the  same  speed,  and  that  all  the  molecules  are  divided 
equally  into  three  groups,  one  group  consisting  of 
molecules  moving  perpendicular  to  the  face  of  the 
piston,  the  other  two  groups  having  motions  parallel 
to  this  face,  but  mutually  perpendicular.  These  two 
last  groups  will  exert  pressure  only  on  the  side  walls 
of  the  cylinder,  and  we  have  to  consider  the  effect 
upon  the  piston  of  the  first  group  only,  whose  motion 
is  perpendicular  to  it. 

Let  A  =  area  of  face  of  piston, 
h  =  height  of  cylinder, 
m  =  mass  of  one  molecule, 
n  =  number  of  molecules  in  unit  space, 
c  =  common  speed  of  all  the  molecules. 

When  one  molecule  hits  the  piston,  if  it  were  able 
to  just  give  up  all  its  motion  to  the  piston  and  itself 
come  to  rest,  it  would  exert  upon  the  piston  an  im- 
pulse exactly  equal  to  its  momentum,  me  \  but  the 
mass  of  the  piston  is  vastly  greater  than  that  of  the 
molecule,  and  the  velocity  produced  in  it  is  vastly 
smaller  than  that  of  the  molecule,  hence  when  the 


IDEAL   GASES.  II 

molecule  has  just  come  to  rest,  if  we  are  to  regard  it 
as  a  smooth,  hard  sphere,  it  is  in  contact  with  tne  pis- 
ton and  strongly  compressed  or  distorted  ;  at  any  rate 
it  is  in  the  very  act  of  rebounding  from  the  piston. 
Now  according  to  the  ordinary  laws  of  reflected  mo- 
tion the  molecule  will  rebound  from  the  piston  with 
the  same  velocity  with  which  it  struck  it,  and  accord- 
ing to  Newton's  third  law,  that  action  and  reaction 
are  equal  and  opposite  in  direction,  there  acts  upon 
the  piston,  still  driving  it  away  from  the  gas,  an  im- 
pulse just  equal  to  the  impulse  upon  the  molecule, 
which  gives  it  again  the  momentum  me.  Hence  the 
total  impulse  upon  the  piston  due  to  a  single  impact 
of  a  single  molecule  is  2mc. 

If  the  molecule  has  just  hit  the  piston,  before  it 
can  hit  it  again  it  has  to  traverse  the  length  of  the 
cylinder  h  and  return,  a  distance  of  2h  in  all,  before 
it  hits  the  piston  again  ;  and  since  it  travels  a  distance 
c  per  second,  it  will  be  able  to  hit  the  piston  cJ2h 
times  per  second.  Since  the  distance  c  is  very  large, 
of  the  order  of  one  mile  in  ordinary  gases,  we  do  not 
need  to  consider  the  possibility  of  one  more  or  one 
less  collision  per  second,  depending  upon  the  exact 
positions  of  the  molecule  at  the  beginning  and  end  01 
the  second.  One  molecule  would  then  in  t  seconds 
hit  the  piston  cJ2h  x  t  times  ;  and  the  sum  of  all  the 
impulses  given  to  the  piston  by  the  orie  molecule  in 
the  time  /  is 

c 
2mc  x  —   X  t 


2h  h 


12  KINETIC   THEORY. 

As  yet  we  have  considered  only  the  effect  of  one 
molecule ;  in  each  unit  of  volume  there  are  n  mole- 
cules, and  in  the  whole  volume  of  the  cylinder,  which 
is  h  x  A,  there  must  be  n  X  hA  molecules.  But  not 
all  of  these  are  effective  in  producing  pressure  upon 
the  piston,  in  fact  we  have  expressly  assumed  that  just 
one  third  of  them,  that  is,  \nhAy  were  so  effective ; 
consequently  the  whole  impulse  upon  the  piston  in 
the  time  t  will  be  the  product 

|  nhA  x  —j —  =  \nmc^At. 

To  produce  equilibrium  of  the  piston  this  impulse 
must  be  opposed  by  a  force  which  will  in  the  same 
time  have  just  the  same  impulse.  If  we  call  this  force 
Fy  we  may  write 

Ft  =  \nnn*At, 

F=  \nni(?A. 

A  force  F  of  this  amount  would  on  the  average  be 
able  to  hold  the  piston  in  position  against  the  repeated 
blows  of  the  molecules.  These  blows  might  seem  to 
cause  a  slight  quivering  of  the  piston,  an  oscillation 
back  and  forth,  but  the  impulse  of  each  blow  is  so 
slight  and  the  number  so  enormous  that  this  oscillation 
can  never  be  actually  observed.  Now  the  force  F 
could  be  exactly  replaced  or  neutralized  by  a  pressure 
/  upon  the  piston  of  such  intensity  that 


IDEAL  GASES.  13 

This  would  give  us 

pA  =  \nrnc21  A 


or 


The  discussion  of  the  validity  of  the  method  by 
which  we  have  derived  this  equation  we  shall  post- 
pone for  a  little.  For  the  present  we  shall  assume 
that  it  is  correct,  and  see  what  are  its  consequences. 

It  appears  immediately  that  the  pressure  is  propor- 
tional to  the  square  of  the  velocities  of  the  molecules, 
that  is,  to  their  kinetic  energy  of  translation.  This 
conclusion  does  not  depend  upon  any  of  the  assump- 
tions made,  but  simply  on  the  two  considerations  that 
the  impulse  of  a  single  impact  is  proportional  to  the 
velocity  of  the  molecule,  and  the  number  of  the  im- 
pacts is  also  proportional  to  the  velocity,  and  hence 
the  total  effect  is  proportional  to  the  square  of  the 
velocity. 

We  may  put  the  equation  in  a  different  form  if  we 
consider  that  since  m  is  the  mass  of  a  single  molecule 
and  n  the  number  of  the  molecules  in  unit  volume, 
the  product  nm  is  simply  the  density  of  the  gas,  which 
we  may  call  p.  Introducing  this  we  have 


which  may  solve  for  c,  getting 


14  KINETIC^  THEORY. 

This  gives  us  a  means  of  computing  the  velocity  of 
the  molecules  of  a  gas  directly  from  a  knowledge  of 
its  pressure  and  density.  If  we  take  hydrogen  as  an 
example,  its  density  at  a  pressure  of  one  million 
dynes  per  sq.  cm.  and  o°  C.  is  given  as  .0000884. 
This  is  very  nearly  atmospheric  pressure,  the  atmos- 
phere being  about  1.013  million  dynes  per  sq.  cm. 
These  figures  give 

c  =     1  3  x  1,000,000 
\      .0000884 

=  1  84,400  cm.  per  second. 

The  formula  shows  that  for  other  gases  at  the  same 
pressure  or  for  the  same  gas  at  different  temperatures 
but  the  same  pressure  the  velocity  c  is  inversely  as 
the  square  root  of  the  density,  hence  we  readily  obtain 
for  oxygen  c  =  46,  100  cm.  per  sec., 
for  nitrogen  ^=49,200  "  "  " 

For  any  volume  vt  calling  the  number  of  molecules 
in  the  space  v 

N  —  vn, 
the  equation 

(i),  p 

becomes 

(2)  pv 


This  is  very  much  like  the  equation 

(3)  PV 


which  describes  the  behavior  of  ideal  gases.     If  we 
regard  the  two  equations  as  identical,  we  conclude  : 


IDEAL   GASES.  15 

First,  that  a  gas  made  up  as  we  have  described  it 
follows  Boyle's  or  Mariotte's  Law. 

Second,  that  such  a  gas  follows  the  law  of  Charles ,  or 
Gay  Lussac,  with  regard  to  change  of  pressure  or  vol- 
ume with  increasing  temperature.  That  is,  a  gas  made 
up  of  an  aggregation  of  small  molecules  with  high 
velocities,  the  molecules  so  small  as  to  occupy  only  a 
negligible  portion  of  the  space  filled  by  the  gas,  would 
exhibit  the  phenomena  of  an  ideal  gas,  which  actual 
gases  closely  approximate. 

In  coming  to  this  conclusion  we  have  really  made 
one  very  important  assumption,  or  perhaps  better,  de- 
finition. We  have,  in  stating  the  identity  of  the  two 
equations 

pv  =  \Nrnc*        pv  =  RT 

stated  that  the  temperature  of  a  gas  is  proportional  to 
the  square  of  the  velocity  of  the  molecules  of  the  gas, 
or  to  the  kinetic  energy  of  the  motion  of  translation 
of  the  molecules.  This  then  really  defines  our  tem- 
perature scale.  We  shall  for  the  present  accept  this 
definition,  and  consider  all  temperatures  measured  on 
the  scale  of  a  thermometer  whose  working  substance 
is  such  an  ideal  gas.  We  shall  consider  this  pro- 
cedure justified  if  its  consequences  are  consistent  with 
well-ascertained  facts. 

Before  examining  further  the  possible  meanings  of 
the  equation  we  shall  consider  the  assumptions  we 
have  made  as  to  the  velocities  of  the  molecules.  That 
these  assumptions  should  be  true  is  inconceivable.  If 


i6 


KINETIC   THEORY. 


the  molecules  of  a  gas  could  be  once  started  to  mov- 
ing in  such  a  way  as  we  have  described,  ia  a  very 
small  fraction  of  a  second  so  many  collisions  would 
have  taken  place  between  the  molecules  whether  of 
the  same  set,  or  of  the  different  sets,  that  molecules 
would  be  moving  in  every  conceivable  direction  and 
with  almost  every  conceivable  speed.  Take  for  in- 
stance such  a  collision  as  that  represented  in  Fig.  2. 

The  molecules  A  and  B 
are  moving  with  equal 
speeds  in  directions  at 
right  angles  to  each  other, 
and  hit  as  shown  in  the 
figure,  so  that  B  gives  up 
all  its  motion  to  A,  which 
had  previously  no  compo- 
nent of  its  motion  in  the 
same  direction,  but  after 
the  collision  has  a  velocity 
the  resultant  of  the  two 
previous  velocities,  and  numerically  equal  to  either 
of  them  multiplied  by  1/2,  that  is,  1.41.  The  velocity' 
of  B  in  this  extreme  case  is  destroyed,  while  that 
of  A  is  made  nearly  half  as  large  again.  B  loses 
all  its  energy,  while,  calling  the  common  speed  of 
each  before  the  collision  r,  the  energy  of  A  af- 
terward is 

i;;/(<:]/2)2  =  me*, 


Fig.  2. 


which  is  just  twice  its  previous  energy.     This  result 


IDEAL  GASES.  I/ 

was  necessary,  for  the  total  energy  before  and  after 
the  collision  must  be  the  same. 

It  is  evidently  impossible  to  follow  the  path  of  each 
molecule,  and  examine  the  conditions  of  all  its  col- 
lisions, so  the  question  arises  whether  there  is  any 
other  method  of  studying  these  actions  which  will 
prove  fruitful  in  results.  We  may  perhaps  obtain  a 
more  definite  conception  of  the  problem  in  the  fol- 
lowing way ;  take  any  convenient  point  as  the  origin 
of  a  system  of  coordinates,  and  from  this  point  draw 
a  line  which  shall  have  the  same  direction  as  that  of 
the  motion  of  some  particular  molecule  and  a  length 
proportional  on  some  convenient  scale  to  its  velocity. 
We  can  think  of  this  line  or  of  its  end  as  represent- 
ing fully  the  velocity  of  the  molecule.  If  we  consider 
all  the  molecules  in  some  small  definite  space,  we 
may  draw  for  each  from  this  same  origin  its  velocity- 
line.  These  velocity-lines  will  then  stick  out  from 
this  origin  in  all  possible  directions,  and  with  a  great 
variety  of  lengths.  We  might  picture  to  ourselves 
the  aggregate  as  an  exaggerated  spherical  hedgehog, 
with  spines  infinitely  numerous  and  of  every  length. 
We  can  conceive  of  no  possible  reason  why  the 
arrangement  of  the  spines  or  velocity-lines  should  be 
different  on  one  side  from  what  it  is  on  any  other ;  we 
must  expect  to  find  just  as  many  of  any  one  length  in 
one  direction  as  in  another. 

The  arrangement  which  we  are  describing  is  one 
which  may  be  called  in  the  strictest  sense  of  that 
term  accidental,  and  is  one  to  which  the  Theory  of 
2 


1 8  KINETIC    THEORY. 

Probabilities  may  be  applied  with  perfect  propriety. 
For  a  complete  exposition  of  this  theory  the  reader  is 
referred  to  more  mathematical  treatises  on  the  Kinetic 
Theory,  or  to  text-books  on  the  Method  of  Least 
Squares.  If  we  pass  any  plane  through  the  origin, 
there  will  be  just  the  same  arrangement  on  each  side 
of  the  plane,  as  if  each  side  were  the  image  of  the. 
other  mirrored  in  the  plane.  If  we  pass  two  planes 
anywhere,  parallel  to  each  other,  but  quite  near 
together,  they  will  contain  between  them  a  thin  layer 
or  sheet  of  space  which  will  have  a  great  many  of 
these  velocity-lines  ending  in  it.  Any  two  such  layers 
of  the  same  thickness  and  distance  from  the  origin 
ought  to  have  just  the  same  number  of  such  lines 
ending  in  them. 

Suppose  that  we  have  drawn  the  velocity-lines  for 
all  the  molecules  in  a  unit  volume,  then  there  will  be 
just  n  of  these  lines.  Now,  how  many  of  these  will 
end  in  a  particular  layer,  such  as  we  have  described  ? 
The  number  will  of  course  be  proportional  to  ny  the 
total  number,  and  to  the  thickness  of  the  layer,  if 
that  be  small.  If  we  call  the  distance  of  the  nearer 
side  of  the  layer  from  the  origin  u,  and  its  thickness 
du,  we  may  write  the  number  of  these  lines 

nf(u)du. 

In  mathematical  terms,  this  is  the  number  of  molecules 
which  have  velocities,  the  Jf-components  of  which  lie 
between  z/^and  u  -f  du. 

The  factor  f(u)du 


IDEAL   GASES.  19 

is  called  the  Probability  that  a  molecule  should  have 
such  a  velocity.  The  function  f(u)  is  a  quantity  in 
some  way  depending  upon  u,  but  whose  form  we  do 
not  as  yet  attempt  to  assign. 

Similarly  nf(y)dv 

is  the  number  of  molecules  having  the  ^components 
of  their  velocities  between  v  and  v  -f  dv,  and 

nf(w)dw 

the  number  of  those  having  ^-components  between  w 
and  w  +dw  ;  ortf(v)dv  and  f(w)dw  are  the  respective 
probabilities  that  a  molecule  should  have  such  veloci- 
ties. We  write  these  functions  all  in  the  same  form, 
because  we  believe  the  law  of  probabilities  must  be 
the  same  in  every  direction. 

Now  the  two  planes  whose  distances  from  the  origin 
are  u  and  u  -f  du  and  the  two  whose  distances  are  v 
and  v  +  dv  intersect  to  form  a  little  rectangular  prism, 
of  infinite  length,  and  of  width  and  thickness  du  and 
dv.  What  is  the  probability  that  a  molecule  has  a 
velocity  whose  line  ends  in  this  prism,  that  is,  in  both 
these  layers  ?  By  the  ordinary  theory  of  probabilities 
it  is  the  product  of  the  separate  probabilites  of  its  end- 
ing in  either  of  the  two  layers,  that  is  it  is 

f(u)f(v)dudv 

and  the  probable  number  of  velocity  lines  ending  in 
the  prism  is 

nf(ii)f(v)dudv. 


20  KINETIC   THEORY. 

The  third  pair  of  planes  cut  this  prism,  forming  a  little 
rectangular  parallelepiped  whose  dimensions  are  dut 
dv,  dw,  and  by  the  same  process  of  reasoning,  the 
probability  that  a  velocity-line  ends  in  this  little  space  is 

f(u)f(^v)f(iu)dudvdw 

and  the  number  of  them  ending  in  this  space  is 
nf(u)f(i>)f(T,v)dudvdw. 

Now  we  know  two  things  very  definitely  about  this 
expression  ;  first  that  the  total  number  of  velocity- 
lines  is  n,  that  is,  that  the  sum  or  integral  of  this  ex- 
pression over  all  space  is  n,  or  taking  out  the  common 
factor  n  that 

(4) 

and  second,  that  the  value  of  the  expression,  that  is, 
the  number  of  lines  ending  in  the  space  dudvdw  de- 
pends only  on  the  size  of  this  space  and  on  its  dis- 
tance from  the  origin,  and  not  on  its  direction.  Now 
the  distance  c  is  given  by  the  equation 

^  =  i?  -f  v2  -f  w* ; 
hence  we  may  write 

f(H)f(v)f(w)  =  #0  =  #««  +  V-  +  K?) 

which  indicates  symbolically  the  fact  we  have  just 
stated.  We  may  then  write  our  expression  for  the 
number  of  velocity-lines  ending  in  the  space  dudvdw 

-f  v2  +  w^dudvdw. 


IDEAL   GASES.  21 

It  is  possible  from  the  facts  which  we  have  just 
stated  to  derive  the  forms  of  the  functions^  and  <j>,  but 
we  shall  take  the  easier  method  of  suggesting  the  form 
of  solution  and  testing  it  to  see  if  it  satisfies  the  condi- 
tions which  we  have  stated.  Professor  J.  Clerk  Max- 
well l  has  suggested  the  solution 

(5)  /(«)-J«r? 

where  e  is  the  base  of  the  natural  system  of  logarithms, 
and  A  and  a  are  constants  to  be  determined,  then 


and  hence  satisfies  the  second  of  our  conditions.  The 
first  will  be  satisfied  by  giving  a  proper  value  to  the 
constant  A.  Inasmuch  as  all  the  molecules  of  the 
gas  have  velocities  whose  JT-components  lie  between 
—  OQ  and  4-  co,  A  must  have  a  value  which  will 
satisfy  the  equation 

rnAe   a2  du  =  n. 
oo 

This  value  is  found2  to  be 

I 


i       _^ 

~—r=*  **> 
ayir 


1  Phil.  Mag.  (4),  19,  p.  22,  1860. 

2  The  equation 

_«2 

rnAe    a2a 
00 


22  KINETIC    THEORY. 

and  the  number  of  velocity  lines  ending  in  the  space 
dudvdw  is 


n 


U*  +  ifl  +  IV* 


e        a*          dudvdw 


We  may  express  these  relations  graphically  by  plot- 
ting the  curve  for  the  equation 

gives 


I*~ 

P     e~*du=i. 

J  —       00 


The  value  of  a  definite  integral  does  not  depend  upon  the  particular 
variable  in  terms  of  which  the  integral  is  written,  hence  we  may  equally 
well  write 


__ 

A   f°   e     *dv=i 

J  —  oo 


_ 
A   P    e     **dw=l. 

J  —  00 

Multiplying  any  two  of  these  together,  for  instance  the  first  two 
A2 


P°      P  f        «z   dudv=i. 

—  00    J  —  00 


We  can  transform  this  expression  into  polar  coordinates,  r  and  8,  by 
writing 

*+f*9H 

and  substituting  for  the  infinitesimal  area  dudv  the  corresponding  ex- 
pression in  polar  coordinates,  rdrd  6.  This  gives 

r2 

a*  rdrd  6=  i. 

The  integration  is  to  be  extended  over  the  whole  area  of  the  plane,  and 
this  is  covered  if  6  vary  from  o  to  2?r,  and  r  from  o  to  oo.  Performing 
the  first  integration  immediately, 


IDEAL   GASES. 


which  is  commonly  known  as  the  Probability  Curve. 
Its  height  at  any  point  represents  the  value  of  f(u) 
corresponding  to  a  particular  value  of  uy  and  if  two 
vertical  lines,  as  AB  and  CD  be  drawn  at  distances 
from  the  origin  u  and  u  -f-  du9  the  area  between  them, 


9 


A  C 


Fig.  3. 


having  the  base  du  and  the  height  f(u)  will  represent 
the  number  of  molecules  having  the  .^-component  of 
their  velocities  between  u  and  u  +  du. 

The  curve  is  evidently  symmetrical  with  reference 
to  the  axis  of  Y.  It  must  be  so,  for  positive  and 
negative  components  are  equally  numerous.  It  is 


f° 


Now  the  differential  of  e    a2  is 


Hence 


2    ~^    , 

—'    "*• 


=irA*a*=I, 


I 
7ras 


24  KINETIC   THEORY. 

highest  in  the  middle,  and  it  can  be  proven  that  this 
corresponds  to  the  actual  distribution  of  velocities. 
It  is  very  low  at  only  a  short  distance,  showing  that 
very  few  molecules  have  excessively  high  speeds. 
The  total  area  between  the  curve  and  the  horizontal 
axis  is  finite,  and  to  correspond  to  the  equation  as 
written  must  be  just  equal  to  unity. 

Inasmuch  as  a  great  part  of  our  interest  is  centered 
upon  the  speeds  of  the  molecules,  and  we  care  com- 
paratively little  about  their  directions  since  the  phe- 
nomena are  the  same  in  all  directions,  it  is  convenient 
to  reduce  this  expression  to  a  form  which  does  not 
contain  coordinates  of  direction,  like  u,  v,  w,  but 
simply  a  coordinate  of  length.  Now  the  finite  factors 
above  may  be  written  in  the  form 

H          —  — 


which  contains  no  reference  to  direction.  Tl.e  factor 
dudvdw  is  simply  the  volume  of  the  small  space  in 
which  the  velocity-lines  under  consideration  end.  We 
•may  transform  this  expression  to  the  corresponding 
form  for  polar  coordinates,  or  we  may  draw  our  con- 
clusions directly.  Consider  the  thin  shell  bounded 
by  spherical  surfaces  of  radii  c  and  c  -f  dc.  All  parts 
of  it  may  be  considered  as  at  the  same  distance  c  from 
the  origin,  and  hence  as  having  the  same  value  for  the 
factor 


IDEAL   GASES.  25 

The  area  of  one  face  of  this  shell  is  4?rr2  and  its 
thickness  dc,  hence  its  volume  is  A^irrdc,  and  the  num- 
ber of  velocity-lines  ending  in  it  is 

n       -4 


That  is  to  say,  there  are  this   number  of  molecules 
which  have  speeds  lying  between  c  and  c  +  dc. 
The  curve  whose  equation  is 


4      - 


(7) 


is  shown  in  Fig.  4.     Mathematically,  this  curve  should 
be  symmetrical,  positive  and  negative  values  of  c  giv- 


Y 

—  — 

/ 

' 

N 

X 

/\ 

\ 

/ 

\ 

x 

/ 

\ 

\ 

7 

sc 

"\ 

K 

5= 

7 

.., 

x 

/ 

^ 

^ 

^ 

X 

" 

^ 

—  - 

B    C 


r.  4. 


ing  the  same  values  of  y.  Physically,  we  consider 
only  positive  speeds,  and  hence  have  to  consider  only 
the  right-hand  half  of  the  curve,  which  is  all  that  is 
shown  in  the  figure.  The  general  characteristics  of 
the  curve  are  sufficiently  shown  by  the  figure,  and  the 
interpretation  of  it  is  similar  to  that  of  Fig.  3.  The 


26  KINETIC   THEORY. 

highest  point  of  the  curve  corresponds  to  the  most 
probable  speed  and  is  found  by  the  ordinary  method 
of  finding  a  maximum  : 

dy          4     f      -C4     2C  --, 


Dividing  out  common  factors, 


c  =  a. 


That  is,  a  is  the  most  probable  speed  of  the  molecules. 
The  ordinate,  7,  for  this  speed  is 


eayir 

That  is,  the  area  of  a  strip  of  the  width  a/io  at  this 
point  is  .0832,  which  is  the  probability  that  a  molecule 
will  have  a  speed  between  ^a  and  |-Ja,  or  the  num- 
ber of  such  molecules  will  be  .0832^,  nearly  one 
twelfth  of  the  whole  number. 

Knowing  the  number  of  molecules  which  have  each 
possible  speed,  we  are  able  to  find  several  interesting 
average  values.  For  instance,  the  average  speed  of 
all  the  molecules  is  found  by  multiplying  the  number 
of  molecules  having  a  certain  speed  into  that  speed, 
doing  this  for  all  possible  speeds,  adding  the  products 
so  formed,  and  dividing  by  the  whole  number  of  mole- 
cules. This  is  the  ordinary  method  of  taking  aver- 
ages. The  analytical  expression  for  this  is 


IDEAL  GASES.  2/ 

i_  p  4n     - 


The  integration  of  this  expression  l  is  somewhat  com- 


The  integration  of  expressions  of  the  form 


may  sometimes  be  helped  by  the  following  expedient  : 


hence  transposing,  dividing  by  2  and  integrating, 

(  8  )  JV-****£r  =  —  \xn-le-x2  _^_  n_^2  r^n- 

That  is,  the  integration  may  be  made  to  depend  upon  the  integration  of 
a  form  like  the  original,  except  that  the  exponent  of  x  is  2  less  than 
before.  Successive  applications  of  this  formula  will,  if  n  be  odd,  make 
the  integration  depend  upon  that  of 

(9) 

or  if  n  be  even,  upon 


which  is  not  directly  integrable  between  finite  limits,  but  is  easily  in- 
tegrable  when  the  limits  are  both  infinite  or  zero,  by  the  device  used  in 
the  last  footnote.  Calling  the  integral  /, 


=  r 
Jo 


or  passing  to  polar  coordinates,  and  making  the  limits  such  as  to  just 
cover  one  quadrant, 


o     Jo  2Jo 


7= 


28  KINETIC   THEORY. 

plicated,  being  accomplished  by  what  is  commonly 
termed  integration  by  parts.  We  give  here  the  result, 
indicating  the  fact  that  it  is  an  average  value  by  a  line 

drawn  over  the  ?. 

2a 
(id]  c  =  —^- 

1/7T 

This  method  of  obtaining  averages  is  perfectly  general 
and  we  may  apply  it  to  other  powers  of  the  speed,  by 
treating  them  as  we  have  c ;  for  instance,  the  average 
value  of  the  square  of  the  speed  is 

Using  these  devices,  the  mean  speed  is 


which  becomes,  letting  c\a  — 


since  jr8*-*8  vanishes  at  both'  the  limits  o  and  oo. 
Similarly  the  mean  square  of  the  speed  is 

4 


the  expressions  ^3<?-*2  and  .rc-*2  vanishing  at  both  the  lower  and  upper 
limits. 


IDEAL   GASES.  29 


1/TT 


I 


We  might  similarly  find  the  average  values  of  £*,  ^4, 
etc.,  but  the  values  which  we  have  deduced  are  the 
only  ones  of  practical  importance. 

We  can  readily  compare  the  relative  magnitudes  of 
the  different  speeds.  The  most  probable  speed  was  a, 
its  square  a2.  The  average  speed  is  somewhat  larger, 
being  2a/l/7r,  its  square  4#2 /TT.  The  average  of  the 
square  of  the  speed,  commonly  spoken  of  as  the 
"mean  square"  of  the  speed,  is  3«2/2. 

These  three  squares  then  are  in  the  ratios  of 

i  :  1.27:  1.5 

or  the  speeds  themselves  in  the  ratios 
I  :  1.128  :  i.22f" 

the  most  probable  speed  being  the  least,  the  square 
root  of  the  "  mean  square  "  the  greatest.  They  are 
represented  in  Fig.  4  by  the  distances  OA,  OB,  OC\ 
respectively.  The  corresponding  values  of 

a3l/7T 

are  found  to  be 

.832     .805      .755 
a   '       a    '       a    ' 


30  KINETIC   THEORY. 

That  is,  the  number  of  molecules  having  speeds  not 
differing  more  than  I  /  20  a  either  way  from  the  three 
speeds  are  .0832/2,  .080572,  .075572,  respectively,  or 
about  i/ 12,  1/12.4,  1/13.25  of  the  whole  number  of 
molecules. 

The  reason  why  these  average  speeds  are  greater 
than  the  "most  probable  speed,"  is  not  so  much  that 
the  higher  speeds  are  more  numerous  as  simply  that 
the  higher  speeds  contribute  so  much  more  to  the 
sum  of  the  products,  and  hence  exert  a  preponderating 
influence.  We  can,  if  necessary,  find  the  number  of 
molecules  whose  speed  does  not  exceed  a  given  value 
c,  by  evaluating  the  integral 

1/7T, 

but  this  involves  very  difficult  and  indirect  methods,1 

1  The  integration  of  the  expression 


-t=f. 

ttV  7T  J0 


depends  upon  the  integration  between  finite  limits  of 

/_yfl 
'        *'<<*> 

which  again  depends  (see  last  footnote)  on 

/-*2, 
,    d*. 

For  small  values  of  x  this  may  be  evaluated  by  substituting  for  e  —  **  the 
series 

Then 
(12) 


IDEAL   GASES.  31 

and  is  perhaps  easiest  done  by  plotting  the  curve  of 
Fig.  4  very  carefully  and  measuring  the  area  between 
it  and  the  horizontal  axis  to  the  left  of  the  ordinate  c. 
It  can  be  shown  that  .4276  of  all  the  molecules  have 
speeds  not  exceeding  a,  the  most  probable  speed  ; 
.5331  of  them  have  speeds  not  exceeding  2a/yvt  the 
average  speed  ;  .6082  do  not  exceed  l/|a,  the  "  mean 
square"  speed;  while  for  1.5,  2,  and  2.5  times  a,  the 
proportions  are  .7877,  .9540,  .9940,  respectively.  It 
can  be  shown  that  not  more  than  one  in  12.5  x  io9 
have  speeds  over  5 a,  and  less  than  one  in  236  x  io40 
over  ioa.  From  a  study  of  these  numbers  one  sees 
what  is  shown  by  simple  inspection  of  the  curve,  that 
the  great  majority  of  these  molecules  have  speeds  not 
much  less,  nor  very  much  greater  than  these  probable 
or  average  speeds  which  we  have  been  discussing. 
^  We  have  previously  attempted  a  computation  of 
the  pressure  exerted  by  a  perfect  gas  upon  the  walls 


For  large  values  of  x  we  may  write 

\Xe    x  dx=  \     e    X  dx — \     e  *  dx 
~xZdx, 


and  the  integration  of  this  last  term  may  be  effected  by  successive  appli- 
cations of  the  formula  developed  in  the  last  note, 


By  the  use  of  one  or  the  other  of  these  two  formulae  the  numbers  given 
in  the  text  can  be  calculated. 


32  KINETIC   THEORY. 

of  the  containing  vessel.  For  convenience  in  effecting 
the  computation  we  made  several  assumptions  which 
we  acknowledge  frankly  were  not  in  accordance  with 
the  probable  facts.  In  particular,  we  assumed  that  all 
the  molecules  had  the  same  speeds,  and  that  all  were 
moving  in  one  or  the  other  of  three  mutually  perpen- 
dicular directions.  Now  while  such  an  arrangement 
might  possibly  exist  for  an  instant  of  time,  it  is  veiy 
improbable,  and  could  not  be  permanent.  The  distri- 
bution of  speeds  and  directions  which  we  have  been 
studying  can  be  shown  to  be  the  most  probable,  and 
to  be  capable  of  permanence.  Assuming  then  that 
the  molecules  have  such  velocities,  how  will  the 

formula 

(i)  p  =  \nnu* 

be  affected  ?  Which  of  the  various  speeds  we  have 
studied  is  to  be  understood  as  the  c  of  this  equation  ? 
Granted  that  our  reasoning  which  leads  to  the  general 
form  of  this  equation  is  right,  have  we  the  right  con- 
stant factor  ?  Let  us  repeat  the  deduction  in  the  light 
of  our  study  of  the  difference  of  velocities.  Suppose 
the  gas  to  be  confined  between  two  plane  parallel 
walls  as  before.  Laterally  it  makes  no  difference 
whether  it  is  bounded  by  a  cylindrical  surface  as  before, 
or  whether  the  parallel  walls  extend  to  an  indefinite 
distance.  We  shall  take  our  system  of  coordinates 
such  that  the  axis  of  X  is  perpendicular  to  these 
walls.  The  speed  of  any  molecule  we  shall  call  c, 
and  the  angle  between  the  direction  of  its  motion  and 


IDEAL   GASES.  33 

the  X  axis  0.  The  component  of  its  velocity  perpen- 
dicular to  the  two  planes  we  can  then  call  either  c  cos  6 
or  u.  The  other  component,  c  sin  0,  parallel  to  the 
planes,  will  not  be  affected  at  all  by  the  impact  with 
the  planes,  and  so  does  not  have  to  be  taken  into 
account.  We  shall  as  before  entirely  disregard  the 
mutual  collisions  of  the  molecules,  because,  while 
these  change  the  velocities  of  individual  molecules,  on 
the  average  they  leave  the  distribution  the  same,  that 
is,  we  assume  that  our  gas  is  in  a  steady  state.  If  as 
before  we  call  the  distance  between  the  two  planes  /i, 
a  molecule  will  travel  between  two  successive  impacts 
against  the  same  plane  a  distance  of  2/1  in  the  direction 
perpendicular  to  the  plane,  or  an  actual  distance 
2/1  /  cos  0.  It  will  hit  the  plane  then  (c  cos  6)1  2/1  or 
u  1  2/1  times  per  second.  The  impulse  given  the  plane 
by  a  single  impact  of  a  single  molecule  will  be,  by 
the  same  reasoning  as  before,  2mc  cos  6,  or  2mu. 
The  total  impulse  from  a  single  molecule  in  a  second 
will  then  be  the  product  of  these,  namely 

me2  cos2  6      mi?  , 
- 


1  The  two  deductions  of  the  equation  of  pressure  given  in  the  text  are 
not  the  only  ones  possible.  Some  forms  of  the  demonstration  depend 
upon  considerations  involving  a  knowledge  of  the  space  occupied  by  the 
molecules.  Others  depend  directly  upon  abstruse  but  general  theorems 
in  dynamics.  The  demonstration  in  the  text  can  be  completed  in  the 
following  manner,  which  is  more  analytical  in  its  form,  but  not  more 
rigid.  We  will  take  into  consideration  a  large  surface,  of  area  s,  so 
large  that  we  can  neglect  the  number  of  molecules  which  pass  in  and 
out  of  the  bounding  cylindrical  surface.  The  volume  we  are  consider- 


34 


KINETIC    THEORY. 


If  we  imagine  a  right  prism  having  bases  of  unit 
area  in  the  two  parallel  planes,  its  volume  will  be 
numerically  equal  to  its  height,  //,  and  the  total  num- 
ber of  molecules  in  it  will  be  nh.  Not  all  these  will 
be  moving  in  such  directions  as  to  hit  the  plane  sur- 
face we  are  considering  within  the  base  of  the  prism, 
but  on  the  average  among  the  myriads  of  molecules 
as  many  will  come  into  the  space  as  go  out,  and  with 

ing  is  hs,  and  the  total  number  of  molecules  in  this  volume  is  nhs.  The 
sum  of  all  the  impulses  in  one  second  due  to  a  single  molecule  of  speed 
c  the  direction  of  whose  motion  makes  an  angle  6  with  the  axis  of  X  is 

me2  cos2  6 
h 

The  number  of  molecules  making  this  angle  6  with  the  axis  of  X  can  be 
found  as  follows.  In  the  accompanying  figure  let  OX  represent  the  di- 

rection of  the  axis  of  X,  and  let  the 
angle  AOX=  0.  All  the  velocity 
lines  drawn  from  O  whose  inclina- 
tions to  OX  lies  between  6  and  d-\-dd 
will  be  comprised  between  the  two 
conical  surfaces  generated  by  the 
rotation  of  OA  and  OB  about  OX 
as  an  axis,  and  the  number  of  them 
will  be  proportional  to  the  solid  an- 
gle subtended  by  the  zone  generated 
by  the  arc  AB.  Now  letting  OA  =  r  the  area  of  this  zone  is 

=  27rr  sin  6rdd  =  27rr2  sin  &/(9. 


The  whole  area  of  the  spherical  surface  described  in  this  rotation  i 
but  since  we  are  concerned  only  with  the  direction  of  the  line,  and  not 
with  the  direction  of  motion  along  that  line,  all  possible  directions  are 
included  by  the  lines  piercing  one  half  the  spherical  surface,  whose  area 
is  27rr2.  The  ratio  of  these  two  areas  then  is 


05) 


27rr2  sin  Odd 


IDEAL   GASES.  35 

similar  velocities,  so  that  while  not  all  the  identical 
molecules  which  are  at  any  one  time  within  this  prism 
strike  its  base,  yet  the  total  number  available  for 
striking  this  base  is  the  same,  namely  nh.  Then  the 
total  impulse  on  the  base  due  to  all  the  molecules  is 
the  sum  of  all  the  impulses  of  all  the  molecules,  or 

A  mi? 

*?~T' 

We  may  take  mj/i  outside  the  sign  of  summation,  and 
remembering  that 


which  gives  the  relative  number  of  molecules  having  the  inclination  of 
their  paths  to  the  axis  of  X  between  6  and  9  -f  dQ.  The  total  number 
of  such  molecules  will  be  then 

nks  sin  Qdd. 

The  number  of  these  having  speeds  between  c  and  c  -\-  dc  could  be  ex- 
pressed according  to  the  formulae  which  we  have  discussed,  but  we  can 
obtain  directly  the  results  of  integrating  these  formulae  by  writing  for  c2 
its  average  value  c2,  which  gives  us  for  the  impulse  arising  from  the 
impacts  upon  the  surface  of  all  the  molecules  whose  directions  lie  be- 
tween the  limits  stated 

me*  cos2  0 

nhs  sin  OdB  X  -  ;  - 
ft 

=  nsniP  cos2  0  sin  0</0, 
and  the  force  which  must  be  applied  to  maintain  equilibrium  is 

-  /»w/2 

F=ps  —  nsmc*J      cos2  6  sin  6dBt 
p  —  nmc*        '2  cos2  6  sin  ddd  =  nm<*  f~— 


which  is  the  same  as  the  result  obtained  in  the  text. 


36  KINETIC   THEORY. 

the  expression  for  the  total  impulse  becomes 


m 


This  being  the  total  impulse  exerted  in  one  second 
upon  unit  area,  is  numerically  equal  to  the  pressure 
which  would  hold  it  in  equilibrium.  Now 


can  be  represented  by  nu2,  that  is,  n  times  the  average 
of  the  square  of  u,  this  being  simply  the  definition  of 
the  average  ;  hence  our  expression  becomes 

p  =  mm?. 

Now  c2  =  u2  +  i?  -f  zu2,  and  summing  for  all  the  mole- 
cules 


nc2  =  mi2  -\-  nv2  -f-  nw2" 
<***j?+tf  +  ^. 

But  we  have  assumed  that  there  is  no  intrinsic  differ- 
ence in  direction  in  the  gas,  hence  we  are  compelled  to 
write 

1?  =  ^  =  ^; 

hence  u2  =  ^ 

and  /  =  nmu2  =  \nm^. 

This  expression  is  identical  with  that  obtained  by 
the  previous  method,  and  hence  we  wish  to  find  why 


IDEAL  GASES.  37 

it  is  that  our  faulty  assumptions  there  led  to  a  correct 
result.  It  is  evident  that  the  r2  of  that  formula  is 
what  we  now  recognize  as  the  "  mean  square  "  of  the 
velocity.  The  method  of  the  deduction  of  the  impulse 
due  to  the  successive  impacts  of  a  single  molecule 
shows  that  it  is  proportional  to  the  kinetic  energy 
associated  with  the  component  of  its  motion  perpen- 
dicular to  the  plane  against  which  the  pressure  is  ex- 
erted. Considerations  of  symmetry  lead  us  to  believe 
that  whatever  may  be  the  motions  of  the  individual 
molecules,  the  total  kinetic  energy  of  translation  of 
the  molecules  of  the  gas  is  equally  distributed  between 
the  three  components  of  the  motion,  that  is,  if  we  write 


)  i  m<*  =      )  i  mtf  +     )  \  mv*  +     ;  i  mat, 


Z)  2 

i  i  i  i 

This  relation,  that  the  kinetic  energies  associated  with 
the  three  components  of  the  motions  of  translation  of 
the  particles  were  equal  to  each  other,  and  each  equal 
to  one  third  the  total  energy  of  translation  of  all  the 
molecules,  was  expressly  stated  in  the  last  deduction. 
In  the  first,  the  assumption  that  all  the  molecules  had 
the  same  speed,  and  that  one  third  the  number  were 
moving  in  a  direction  parallel  to  each  of  the  three 
principal  axes  made  a  correct  distribution  of  the 
energy,  although  in  a  manner  entirely  arbitrary.  But 
since  the  result  obtained  depended  only  on  the  distri- 
bution of  the  energy,  and  not  upon  the  device  by 


38  KINETIC   THEORY. 

which    this  was  accomplished,   the   results  obtained 
were  trustworthy. 

So  ir  we  have  restricted  ourselves  to  a  study  of 
the  behavior  of  a  gas  all  of  whose  molecules  were 
exactly  alike  ;  now  we  shall  ask  what  is  the  behavior 
of  a  mixture  of  different  kinds  of  gases.  We  shall 
assume  that  we  have  in  the  space  that  we  are  con- 
sidering several  classes  of  molecules,  which  we  shall 
distinguish  by  the  subscripts  I,  2,  etc.  Then  in  unit 
volume  the  numbers  of  molecules  of  each  kind  will  be 
nlf  n2,  -  •  •  respectively  ;  the  masses  of  single  molecules 
mlf  mv  etc.  Now  from  our  last  deduction  of  the 
pressure  exerted  by  a  gas  on  a  plane  surface,  it 
appears  that 


and  the  deduction  of  the  formula  in  this  form  does 
not  depend  upon  any  assumption  as  to  the  uniformity 
of  either  the  masses  or  speeds  of  the  molecules  ;  hence 
if  there  are  several  sorts,  provided  in  each  sort  there 
is  a  thorough  distribution  of  the  velocities,  if 

n  =  «!  +  n^  +  «3  -f  etc. 
we  may  write 


etc. 


That  is,  the  total  pressure  exerted  by  a  mixture  of 
several  gases  upon  the  walls  of  the  receptacle  contain- 


IDEAL   GASES.  39 

ing  them  is  the  sum  of  the  separate  pressures  which 
each  would  exert  if  it  were  occupying  the  same  space 
alone.  This  is  often  called  DALTON'S  LAW.  ; 

We  next  wish  to  find  the  result  of  the  mutual 
collisions  between  molecules  of  two  different  kinds. 

If  we  draw  the  line  joining  the  centers  of  the  two 
colliding  molecules  just  at  the  instant  of  collision,  the 
mutual  forces  of  the  collision  will  act  along  this  line, 
and  the  components  of  the  velocities  of  the  molecules 
in  this  direction  will  suffer  change,  while  the  com- 
ponents perpendicular  to  this  direction  will  not  be 
changed,  and  will  not  need  to  be  taken  into  account. 
Since  the  molecules  are  perfectly  elastic,  we  have  not 
simply  the  sum  of  the  momenta  of  the  two  molecules 
along  this  line  the  same  after  impact  as  before,  but 
also  the  sum  of  the  kinetic  energies  of  translation  of 
the  two  molecules  will  remain  constant,  none  of  it 
having  the  opportunity  to  degenerate  into  motion  of 
smaller  parts.  If  we  represent  these  component  veloci- 
ties by  /j  and  p2  before  the  collision  and  Pl  and  P2 
after  it,  we  have  for  a  single  collision 


which  are  sufficient  to  determine  P^  and  P2.     From 
the  first  equation  we  get 

/V-A+JCA-^ 

m\ 

Substituting  this  in  the  second  and  solving  for  P2  we 
get 


40  KINETIC   THEORY. 


^ 

and  hence 

p  __ 


The  second  pair  of  values  indicates  simply  that  the 
equations  are  satisfied  if  the  molecules  neither  of  them 
change  their  motion  ;  but  this  case  we  are  expressly 
excluding  from  our  consideration,  hence  the  first  pair 
of  values  is  that  in  which  we  are  interested.  Using 
these,  we  find  that  the  difference  of  the  kinetic  energies 
of  the  two  molecules  after  the  collision  is 


im  **\  +    »V»2 


Now  this  equation  applies  simply  to  a  single  collision 
of  a  single  pair  of  molecules.  In  the  case  of  such  a 
mixture  as  we  are  considering  there  will  be  an  ex- 
ceedingly large  number  of  such  collisions  and  what 
we  are  interested  in  most  is  the  sum  total  of  effect,  or 
the  average  effect.  It  does  not  appear  from  the  above 
expression  whether  the  difference  in  the  kinetic  en- 
ergies of  the  two  molecules  is  increased  or  decreased. 
With  regard  to  the  effect  upon  the  gases  in  general, 
we  can  arrive  at  more  definite  conclusions. 

The  second  term  of  the  second  member  of  the  equa- 
tion will  be  the  same  for  eveiy  collision  of  the  sort  we 
are  considering  except  for  the  factor  pl  /2.  If  we  con- 


IDEAL  GASES.  41 

sider  all  the  cases  in  which  p^  and  p2  have  given  mag- 
nitudes, we  believe  that  there  will  be  just  as  many 
cases  where  they  have  the  same  signs  as  where  they 
have  the  opposite  signs  ;  hence  adding  all  these  to- 
gether, all  the  terms  of  this  form  will  cancel  each 
other,  and  we  need  only  to  consider  the  effect  of  the 
first  term. 

The  second  factor  of  the  first  term  is  simply  the 
difference  of  the  energies  of  the  molecules  before  the 
collision,  hence  we  conclude  that  the  average  difference 
of  their  energies  after  their  collision  is  greater  or  less 
than  the  average  difference  before,  according  as  the 
absolute  value  of  the  factor 


is  greater  or  less  than  unity.  The  expression  is  per 
fectly  symmetrical  with  regard  to  ml  and  mv  As 
suming  that 


fa+itf-     -If 

this  expression  may  be  reduced  to 

W», 

K+^2)2 

^mjn^  <  m*  -f  m*  -f  2mlm 
o  <  m*  -f-  m£  —  2mlm2y 
o  <  (;^1  _  mtf, 


42  KINETIC   THEORY. 

which  is  true  unless 

ml  =  m2 ; 
hence  the  factor 

8m 


12 

9  "~~    A 


(m,  + 

is  less  than  unity  whether  ml  or  m2  be  greater,  and 
the  difference  between  the  average  kinetic  energies  of 
the  two  sets  of  molecules  tends  to  decrease  with  every 
collision.  This  very  important  theorem  is  due  to 
Professor  J.  Clerk  Maxwell.1  We  conclude  from  it  that 
when  a  mixture  of  gases  is  in  equilibrium,  the  average 
kinetic  energies  of  each  of  the  kinds  of  molecules  will 
be  the  same. 

Now  we  know  that  the  physical  result  of  intimate 
contact  and  mixture  is  equality  of  temperature,  and 
we  have  previously  been  led  to  believe  that  the  tem- 
perature of  a  single  gas  is  proportional  to  the  mean 
square  of  the  velocity  of  its  molecules,  or  to  their 
mean  kinetic  energy,  hence  we  state  : 

Two  gases  are  at  the  same  temperature  when  their 
molecules  have  the  same  mean  kinetic  energy  ;  and  the 
temperature  of  a  gas  is  proportional  to  the  mean  kinetic 
energy  of  translation  of  its  molecules. 

If  different  gases  are  at  the  same  temperature  and 
pressure,  we  may  write 


and 

*  Phil.  Mag.  (4),  19,  p.  25,  1860. 


IDEAL  GASES.  43 

from  which  we  conclude  that 


or,  equal  volumes  of  different  gases  under  the  same 
pressure  and  at  the  same  temperature  contain  the  same 
number  of  molecules.  This  result  is  known  as  Avo- 
GADRO'S  LAW  and  has  been  reached  independently 
from  purely  physical  and  chemical  considerations, 

Thermodynamics  of  an  Ideal  Gas.  —  The  laws  of 
thermodynamics  are  deduced  in  a  manner  which  is 
entirely  independent  of  any  assumptions  as  to  the 
exact  mechanism  of  that  form  of  energy  which  we  call 
heat,  but  can  evidently  be  used  equally  well  for  the 
study  of  heat  phenomena  in  cases  where  the  mecha- 
nism is  explicitly  stated.  The  principle  of  the  con- 
servation of  energy  is  often  called  the  first  law  of 
thermodynamics  and  stated  as  follows  : 

If  any  quantity  of  heat  is  given  to  an  object  or  a 
system  of  objects,  the  sum  of  its  total  effects  in  in- 
creasing the  internal  energy  and  in  causing  the  sys- 
tem to  do  work  against  external  forces  is  proportional, 
or  if  measured  in  proper  units,  equal  to  the  amount 
of  heat  so  given.  This  law  is  often  briefly  expressed 
symbolically  by  the  equation 


where  dQ  represents  the  heat  given  to  the  substance, 
dU  the  increase  of  its  intrinsic  energy,  and  dW  the 
external  work  done  by  it.  As  the  external  work  often 
consists  in  increasing  the  volume  of  the  substance 


44  KINETIC   THEORY. 

against  external  pressure,  the  term  dW  is  often  re- 
placed by  pdv,  giving  us  the  less  general  form 

dQ  =  dU  +  pdv. 

In  the  case  of  such  an  ideal  gas  as  we  have  just  been 
considering  the  energy  of  a  definite  amount,  containing 
N  molecule,  is  evidently  ^Nmc2,  and  further,  since  we 
have  seen  that  the  temperature  is  proportional  to  the 
energy,  we  may  write  this  U  =  ^Nmc1  =  CT  where  C 
is  some  constant.  The  first  law  then  becomes  for 
such  a  gas 

dQ=  CdT  +  pdv. 

It  appears  immediately  from  this   equation  that  this 

constant  C  is  the  amount  of  heat  required  to  increase 

the  temperature  one  degree  when  the  volume  is  kept 

constant,  or  the   specific   heat    at    constant   volume, 

hence  we  will  indicate  this  by  the  subscript  vt  and 

write 

(i  6)  dQ=  CdT  +  pdv. 

This  will  be  recognized  as  a  familiar  form  of  the 
first  law  as  applied  to  ideal  gases.  From  the  state- 
ments which  we  have  previously  made,  that  the  tem- 
perature is  proportional  to  the  mean  kinetic  energy  of 
the  molecules,  it  follows  immediately  that  increasing 
the  volume  of  the  gas  without  changing  its  temperature 
does  not  change  its  energy,  that  is,  that  dUjdv  —  o. 
This  result  which  follows  immediately  from  the  ki- 
netic theory,  has  been  found  by  very  careful  experi- 
ments to  hold  true  for  actual  gases  just  in  proportion 


IDEAL   GASES.  45 

as  they  conform  to  the  equation  of  ideal  gases, 
pv  =  RT,  or  may  be  deduced  for  gases  conforming 
to  this  equation  by  means  of  the  second  law,  which 
will  be  referred  to  later. 

If,  on  the  other  hand,  the  gas  be  allowed  to  expand, 
but  kept  under  constant  pressure,  the  external  work 
will  be 


If  the  equation  of  the  gas  be  written 

(2)  pv  =  \Nm? 

this  becomes 


The  increase  in  intrinsic  energy  in  increasing  the 
speeds  by  the  same  amount  will  be 

\Nm~^  -  \Nn^  =  \Nm  (tf  -^2). 

This  last  is  the  energy  absorbed  in  heating  the  gas 
at  constant  volume  simply,  while  the  sum  of  the  two, 
|  NM  (c£  —  cf),  is  the  amount  absorbed  in  heating  it  at 
constant  pressure.  Hence  if  we  designate  the  specific 
heat  at  constant  pressure  by  Ct  we  have  the  relation 


C 


This  "ratio  of  the  two  specific  heats"  is  a   quantity 
which  can  be  determined  directly  by  experiment;  and 


46  KINETIC   THEORY. 

is  found  to  have  different  values  for  different  gases,  but 
for  none  to  exceed  this  value.  The  causes  of  variation 
will  be  discussed  later,  but  evidently  such  an  ideal 
gas  as  we  have  been  studying  is  to  be  considered  as 
monatomic,  and  for  mercury  vapor,  which  is  on 
account  of  its  density  regarded  as  monatomic,  the 
value  of  this  ratio  is  found  to  be  1.666. 
By  comparison  with  the  equation 

(3)  PV  =  RT 

it  can  be  easily  shown  that 

07)  Cp=C,  +  R 

and  that  the  first  law  may  be  written  in  the  other  two 

forms 

(i  8)  dQ=CpdT-vdp, 

(19)  dQ^-j^pdv  +  ^vdp. 

Incidentally  since 


and  since  for  different  gases  at  the  same  temperature  the 
mean  kinetic  energy  of  the  molecules  is  the  same, 


N         T  ' 

is  independent  of  the  kind  of  molecule,  or  otherwise 
stated,  The  specific  heat  of  the  molecule  is  the  same  for  all 
gases  ;  or,  otherwise,  the  specific  heats  of  gases  are 
inversely  as  their  njolecular  weights,  or  yet  again, 


IDEAL  GASES.  47 

equal  volumes  of  gases  have  equal  capacities  for  heat. 
This  conclusion  is  to  be  taken  subject  to  the  limita- 
tions which  we  have  just  stated,  namely,  that  it  is 
based  upon  assumptions  which  only  apply  to  especially 
simple  monatomic  gases. 

In  treatises  on  thermodynamics  it  is  shown  that 
dQ  is  not  a  perfect  differential,  that  is,  mathematically 
speaking,  if  two  different  states  of  a  gas  are  designated 

(2 
dQ  may  have  very  different 
. 
values,  according  to  the  manner  in  which  the  gas  is 

made  to  pass  from  the  state  I  to  the  state  2.  Physi- 
cally, the  amount  of  heat  the  gas  will  absorb  in  pass- 
ing from  the  state  I  to  the  state  2  depends  upon  the 
manner  in  which  it  is  made  so  to  pass.  We  apply  to 
such  a  passage  from  one  state  to  another  the  term 
Transformation.  There  are  two  particularly  important 
types  of  transformations  which  are  called  reversible,  the 
isothermal  transformation,  in  which  the  temperature 
of  the  body  is  kept  constant,  and  the  adiabatic  trans- 
formation in  which  no  heat  is  allowed  to  enter  or  leave 
a  body.  Any  series  of  transformations  at  the  end  of 
which  a  body  is  in  exactly  the  same  condition  in  which 
it  was  at  the  beginning  of  the  series  is  called  a  Cycle. 
A  reversible  cycle  is  one  made  up  entirely  of  rever- 
sible transformations.  The  simplest  reversible  cycle  is 
Carnofs  reversible  cycle,  which  is  made  up  of  two 
isothermal  and  two  adiabatic  transformations. 

In  the  accompanying  diagram  if  we  represent  the 
volume  of  a  given  body  of  gas  by  the  abscissa,  and  its 


48 


KINETIC   THEORY. 


pressure  by  the  ordinate  of  a  point,  since  these  two 
also  determine  its  temperature,  we  can  regard  the 
point  as  determining  or  representing  completely  the 
state  of  the  body.  A  transformation  will  be  repre- 
sented on  this  diagram  by  a  line,  one  in  which  the 


Fig.  6. 

pressure  is  kept  constant  by  a  horizontal  line,  if  the 
volume  is  kept  constant  by  a  vertical  line,  or  if  the 
temperature  is  kept  constant  by  the  curve  whose 
equation  is  Boyle's  Law,  pv  —  const.,  that  is,  an  equi- 
lateral hyperbola,  with  the  axes  as  asymptotes.  To 
find  the  equation  of  an  adiabatic  transformation  we 
may  take  any  of  the  forms  of  the  first  law,  but  will 
select  the  one  containing  the  two  coordinates  /  and  vt 


(19) 


IDEAL   GASES.  49 

The  definition  of  an  adiabatic  transformation  is  that 
no  heat  is  allowed  to  enter  or  leave  the  substance, 
that  is 

dQ**o. 
Hence 

C  C 


Cp  dv      dp  _ 

C 

— f  log  v  -f  log  p  =  const, 

or 

Cp 

(2O)  pv  c'»  =  const, 

These  adiabatic  curves  are  very  much  like  the  isother- 
mals,  except  that  they  are  everywhere  steeper.  In 
Fig.  6  A  BCD  represents  a  Carnot's  Cycle,  of  which 
AB  and  DC  represent  isothermal  and  AD  and  BC 
adiabatic  transformations.  Suppose  the  gas  to  be 
brought  from  the  state  A  to  the  state  C  first  by  the 
transformations  AB  and  BC,  and  then  by  the  trans- 
formations AD  and  DC.  The  amount  of  heat  ab- 
sorbed in  the  first  transformation  is 

XB  f*B  f*B 

dQ  =        CdT  +       pdv, 
JA  JA 

in  which,  since  the  temperature  is  constant,  dT,  and 
hence 


CB 

I     Cc 

JA 


CdT 

'A 


SO  KINETIC    THEORY. 

vanishes,  and  the  amount  of  heat  required  is  equal  to 
the  amount  of  work  done,  that  is 


f^=i 


BRT 


QRC  is  explicitly  stated  to  be  zero,  since  BC  is  an  adia- 
batic  transformation.  Similarly  in  the  transformation 
ADC 


and 


These  two  quantities  of  heat  are  not  equal,  and  by 
algebraic  substitutions  it  is  easy  to  show  that  the 
amount  of  work  done  in  the  adiabatic  transformations 
BC  and  AD  is  the  same,  and  hence  the  difference  be- 
tween QAB  and  QDC  is  equal  to  the  difference  between 
the  amounts  of  work  done  by  the  gas  in  the  two 
transformations,  or  to  the  area  enclosed  by  the  figure 
A  BCD.  But  according  to  the  theory  of  differential 
equations  there  should  be  an  integrating  factor  for  the 
equation,  and  I  /  T  is  such  a  factor,  for  multiplying 
both  members  of  the  equation 

dQ=*  CvdT  +  pdv 
by  1/7",  and  remembering  that/  =  RTjv  we  have 

dQ         dT        dv 
T  =  C«-T  +  R^' 

of  which  the  second  member  is  an  exact  differential, 
of  the  quantity 

Cv  log  r+  R  \ogv. 


IDEAL  GASES.  51 

We  may  therefore  write 

dQ      ^ 
-y-  =  *S» 

(21)  ,S  =  Cv  log  T  +  R  log  v  -f  const. 

Now  this  quantity  S,  which  we  here  meet  simply  as  a 
quantity  which  satisfies  a  certain  differential  equation, 
is  called  the  entropy  of  the  gas.  We  may  get  a 
physical  conception  of  its  meaning  in  the  following 
manner : 

The  condition  for  an  adiabatic  transformation  is 

dQ=o, 
if  this  be  true  then 

^=o, 

and  in  an  adiabatic  transformation  the  entropy  of  the 
gas  is  not  changed,  or  in  other  words,  the  transforma- 
tion is  isentropic.  Hence  we  can  think  of  the  entropy 
of  a  substance  as  the  property  which  is  not  changed 
when  the  substance  is  compressed  or  expanded  with- 
out allowing  heat  to  enter  or  leave  it. 
The  differential  equation  for  the  entropy 

TdS=dQ 

gives  us,  by  analogy  with  the  equation, 
pdv  =  dW, 

a  suggestion  as  to  its  nature.  In  both  equations  the 
right-hand  side  represents  energy,  the  first  in  the  form 
of  heat,  the  second  in  the  form  of  work  against 
external  forces.  The  left-hand  side  is  of  the  same 


52  KINETIC   THEORY. 

form  in  both  equations,  consisting  of  what  may  be 
called  an  intensity  factor,  T  or  /,  and  a  quantity  factor, 
dS,  or  dv,  the  differential  of  the  coordinate  in  which 
change  is  experienced.  Entropy  is  then  a  quantity 
factor  rather  than  an  intensity  factor,  and  bears  the 
same  relation  to  temperature  and  heat  which  the  volume 
of  a  substance  does  to  its  pressure  and  work.  We  do 
not  know  the  dimensions  of  either  temperature  or 
entropy,  but  only  those  of  their  product,  heat.  In  this 
respect  they  are  like  the  electrical  units,  whose  dimen- 
sions are  made  apparently  definite  only  by  perfectly 
arbitrary,  although  convenient,  assumptions. 

Other  expressions  for  the  entropy  may  be  deduced 
either  by  substituting  the  values  of  v  or  T  from  the 
equation,  or  directly. 

The  form  which  we  have  deduced 

(21)  5=  £7  log  T+  Rlogv  -f  const, 
easily  reduces  to  the  form 

(22)  S  =  R  log  (T~*v)  +  const. 

which  for  the  monatomic  gases  we  have  been  studying 
takes  the  simple  form 

5  =  R  log  (T1  v)  -f  const. 
The  equation 

dQ  =  TdS 

or  the  special  form  for  a  reversible  cycle 


IDEAL  GASES.  53 

which  means  that  the  integral  of  the  function  dQjT 
taken  completely  about  such  a  cycle  vanishes,  or  in 
other  words  that  the  entropy,  of  which  dQj  T  is  the 
exact  differential,  depends  only  upon  the  state  of  the 
substance,  and  not  on  its  history,  is  a  mathematical 
statement  of  the  Second  Law  of  Thermodynamics. 
The  particularly  simple  form  of  these  equations  de- 
pends upon  our  happy  choice  of  a  thermometer 
scale,  and  hence  upon  the  properties  of  the  ideal 
gas.  A  qualitative  statement  of  this  law  can  be 
given  in  several  forms.  Clausius  states  it  as  fol- 
lows : 

It  is  impossible  for  a  self-acting  machine  unaided  by 
any  external  agency,  to  convey  heat  from  one  body  to 
another  at  a  higher  temperature. 

Lord  Kelvin  gives  it  the  slightg^lj  different 
form: 

It  is  impossible,  by  means  of  inanimate  material 
agency,  to  derive  mechanical  effect  from  any  portion  of 
matter  by  cooling  it  beloiv  the  temperature  of  the  coldest 
of  the  surrounding  objects. 

The  validity  of  this  Second  Law  is  a  matter  of 
experience,  and  is  not  restricted  to  any  particular  sub- 
stances. The  reason  seems  to  be  that  we  are  not  able 
to  deal  individually  with  the  motions  of  molecules, 
and  discriminate  between  those  with  more  and  those 
with  less  energy,  but  have  to  deal  with  them  in  a 
lump.  Hence  it  is  that  our  treatment  of  the  Kinetic 
Theory,  dealing  as  it  does  with  averages,  presents  the 
Second  Law  as  a  matter  of  course. 


54  KINETIC   THEORY. 

This  is  illustrated  by  the  conception  sometimes 
spoken  of  as  "Maxwell's  Demon-Engine"  which  is 
described  by  him  as  follows  : * 

"  But  if  we  conceive  a  being  whose  faculties  are  so 
sharpened  that  he  can  follow  every  molecule  in  its  course , 
such  a  being,  whose  attributes  are  still  as  essentially 
finite  as  our  own,  would  be  able  to  do  what  is  at  present 
impossible  to  us.  For  we  have  seen  that  the  molecules 
in  a  vessel  full  of  air  at  a  uniform  temperature  are 
moving  with  velocities  by  no  means  uniform,  though  the 
mean  velocity  of  any  great  number  of  them,  arbitrarily 
selected,  is  almost  exactly  uniform.  Now  let  us  suppose 
that  such  a  vessel  is  divided  into  two  portions,  A  and  B, 
by  a  division  in  which  there  is  a  small  hole,  and  that  a 
being,  who  can  see  the  individual  molecules,  opens  and 
closes  this  hole,  so  as  to  allow  only  the  swifter  molecules 
to  pass  from  A  to  B,  and  only  the  slower  ones  pass  from 
B  to  A.  He  will  thus,  without  expenditure  of  work, 
raise  the  temperature  of  B  and  leaver  that  of  A,  in  con- 
tradiction to  the  second  law  of  thermodynamics" 

1  Maxwell,  "Theory  of  Heat,"  p.  328. 


CHAPTER  III. 
GASES  WHOSE  MOLECULES  HAVE  DIMENSIONS. 

Mean  Free  Path. — Hitherto  we  have  entirely  dis- 
regarded the  space  actually  occupied  by  the  molecules 
themselves,  and  in  consequence  of  this  have  paid  no 
attention  to  the  collisions  between  different  molecules, 
except  for  one  theorem,  but  have  contented  ourselves 
with  the  assurance  that  as  we  were  considering  only 
stable  states,  the  countless  collisions  which  we  believe 
did  take  place  still  left  the  general  distribution  of  the 
molecules,  and  of  their  speeds  and  their  directions  the 
same  as  they  were  before.  Furthermore,  it  is  only  as 
the  molecules  have  some  extension  that  they  are  able 
to  hit  one  another.  But  we  wish  now  to  enquire 
more  minutely  how  far,  on  the  average,  a  molecule 
travels  after  hitting  one  molecule  before  hitting  the 
next,  how  often  it  hits  other  molecules,  and  how  many 
collisions  per  second  there  are  in  any  given  space. 
The  answers  to  these  questions  we  must  approach 
step  by  step,  overcoming  first  one  difficulty,  then 
another. 

For  convenience  we  may  at  first  regard  all  the 
molecules  except  one  as  fixed  in  their  positions,  and 
shall  ask  the  question,  how  far  must  that  molecule 
move  before  it  strikes  against  one  of  the  fixed  mole- 
cules ;  or  we  may  regard  the  molecule  we  are  con- 

55 


56  KINETIC   THEORY. 

sidering  as  fixed  in  space,  and  all  the  others,  still 
keeping  their  positions  relative  to  each  other  un- 
changed, moving  in  a  direction  just  opposite  to  that 
the  single  molecule  had,  but  with  a  speed  just  equal  to 
that  which  it  had  possessed.  We  can  then  state  our 
problem  in  an  entirely  different  way  :  What  is  the  prob- 
ability that  some  portion  of  the  surface  of  some  par- 
ticular molecule  will  hit  the  one  fixed  molecule  within 
a  certain  time  ?  If  the  original  speed  of  the  molecule 
was  c,  then  in  the  short  time  dt  everything  in  our  mov- 
ing system  will  have  traveled  a  distance  cdt,  including 
this  particular  molecule,  which  will  have  described  a 
little  prism  whose  slant  height  will  be  cdt,  and  whose 
cross-section  will  be  the  cross-section  of  the  molecule, 
which  we  shall  call  s,  hence  its  volume  will  be  scdt. 
Now  if  this  surface  is  to  hit  the  fixed  molecule,  the 
molecule  must  lie  within  this  little  prism,  hence  the 
probability  that  it  will  hit  the  other  molecule  within 
the  time  dt  is  the  probability  that  it  will  lie  within  the 
little  volume  generated  by  it.  But  if  nothing  further 
is  specified  as  to  the  location  of  the  molecule  than 
simply  that  it  is  somewhere  in  a  certain  large  vojume 
F,  which  includes  the  little  prism,  the  probability  that 
it  lies  in  the  small  volume  is  simply  the  ratio  of  the 
two  volumes,  or 


and  the  probability  that  it  will  hit  some  one  of  the  n 
molecules  in  a  unit  volume  is 

nscdtj  V. 


MOLECULES   HAVING   DIMENSIONS.  57 

The  probability  of  its  hitting  some  portion  of  this 
surface  ns  is  evidently  proportional  to  the  time,  and 
hence  if  the  time  is  taken  long  enough  it  is  sure  to 
hit.  The  mathematical  expression  for  certainty  is 
unity,  and  if  we  call  the  time  in  which  our  molecule 
is,  on  the  average,  just  sure  to  hit  the  surface  ns,  r, 
we  have 

nscT 

^=I 

and 

V 

r  =  —  ; 
nsc 

r  is  then  the  average  time  within  which  the  point  will 
hit  some  part  of  the  surface,  and 

I       nsc 


expresses  the  number  of  such  collisions  per  second. 
Since  the  molecule  is  moving  with  a  speed  c,  the  dis- 
tance it  will  travel  in  the  time  r  is 


ns 

which  is  then  the  mean  free  path  of  the  molecule,  that 
is,  the  average  distance  a  molecule  travels  between 
two  successive  collisions. 

But  this  solution  is  only  formal,  and  gives  us  no 
insight  into  the  real  occurrences.  We  may,  however, 
make  use  of  our  assumption  that  the  molecules  are 
smooth  hard  spheres,  then  calling  the  diameter  of 


5  8  .  KINETIC   THEORY. 

these  spheres  cr,  whenever  two  spheres  are  in  contact 
their  centers  are  at  a  distance  a  from  each  other.  If 
we  are  considering  one  molecule  as  moving,  and  all 
the  rest  as  fixed  in  their  relative  positions,  we  may 
regard  the  moving  molecule  as  simply  a  point,  located 
at  the  center  of  the  actual  molecule,  and  construct 
about  all  the  other  molecules  spherical  surfaces  of 
radius  cr  with  their  centers  at  the  centers  of  the  mole- 
cules, which  we  may  call  their  spheres  of  action. 
Then  whenever  the  moving  point  touches  one  of  these 
spherical  surfaces  we  have  the  conditions  for  a  col- 
lision, since  the  centers  of  the  two  molecules  are  sep- 
arated by  a  distance  cr.  If  then  we  consider  all  the 
fixed  molecules  replaced  by  these  spheres  of  action, 
the  little  prisms  we  have  imagined  will  be  generated 
by  these  spheres,  and  their  cross-section,  which  we 
have  called  s,  will  be  the  area  Trcr2  of  a  central  section 
of  one  of  these  spheres,1  and  the  volume  of  one  of  the 

1 A  more  analytical  form  of  demonstration  is  this  :  We  are  to  find 
the  probability  that  a  portion  of  the  surface  of  a  moving  molecule  of  area 
ds  will  hit  the  fixed  molecule.  In  the  time  dt  it  will  generate  a  prism 
of  slant  height  cdt,  and  cross-section  ds  cos  6  where  0  is  the  angle  be- 
tween the  normal  to  the  surface  and  the  direction  of  motion.  This 
makes  the  volume  of  the  cylinder  ds  cos  0  cdt.  As  we  have  no  reason 
for  assigning  any  particular  direction  to  the  motion  of  the  molecule,  we 
seek  an  average  value  of  6.  We  have  seen  (p.  34,  (15))  that  if  all  the 
velocity  lines  belonging  to  n  molecules  be  drawn,  if  they  be  uniformly 
distributed  in  every  direction,  the  number  of  them  having  directions  be- 
tween 6  and  6  -f-  dQ  is 

n  •  2ir  sin  BdB      n  sin  6dQ 

W  ~2~ 

We  find  the  average  value  of  cos  6  by  multiplying  by  this  number,  inte- 


MOLECULES   HAVING   DIMENSIONS.  59 

prisms,  or  cylinders,  will  be  ira^cdt.     The  volume  of 
all  the  spheres  of  action  of  the  molecules  in  a  unit 

volume  is 

n  . 


Hence  the  remaining  free  volume  in  which  our  point 
can  move  is 

V=  I  — 


Substituting  these  values  we  have 

V  _i- 

ns  Trna 

or  if  the  volume  of  these  spheres  of  action  is  so  small 
that  we  can  disregard  it  in  comparison  to  the  total 
volume  of  the  gas 


and  the  number  of  collisions  per  second  of  this  mole- 
cule is 


grating  from  o  to  77/2,  since  negative  values  of  cos  6  do  not  correspond 
to  possible  collisions,  and  dividing  by  n  which  gives 


I  r*ft  .«sin0<#  .  rsin*0 
-  /  cos  0  —  —  =  i  — 
nJo  2  L  2 


2       o 

that  is,  the  average  cross-section  of  the  prisms  is  |  the  exposed  area  of 
the  molecule.  This  makes  the  volume  of  the  average  little  prism  \dscdt. 
Integrating  this  over  the  sphere  of  action, 


J  \ds  — 


that  is,  the  effective  area  of  this  sphere  is  \  its  total  area,  or  the  area  of 
a  central  section,  which  we  have  used  in  our  previous  demonstration. 


60  KINETIC   THEORY. 

The  next  question  which  arises  is,  what  speed  is  to  be 
understood  by  c  in  this  formula.  We  have  implicitly 
considered  that  it  was  the  average  speed,  c,  but  only 
by  assuming  all  other  molecules  at  rest.  A  more  ac- 
curate result  would  be  given  by  considering  the  rela- 
tive speed,  r,  of  the  moving  molecule,  and  if  its  aver- 
age value  be  r,  then  the  number  of  collisions  per 
second  of  the  molecule  will  be 


(23)  P 

But  the  distance  the  molecule  travels  between  two 
successive  collisions  is  not  determined  by  r,  its  relative 
velocity,  but  by  c  its  actual  velocity,  hence  if  we  have 
the  number  of  collisions  per  second  just  found  above, 
and  the  molecule  is  moving  with  the  average  speed 
cy  the  average  distance  between  collisions,  or  the  mean 
free  path  is  \  c 

~  7rncr2r 

We  must  then  find  the  value  of  the  ratio  -.     This 

r 

may  be  found  approximately  in  the  following  manner  :  l 

1  The  approximate  demonstration  given  in  the  text  is  due  to  Clausius 
(  "  Kinetische  Theorie  der  Case,"  p.  46).  If  we  assume  that  the  mole- 
cules have  not  all  the  same  speeds,  but  that  they  have  Maxwell's  dis- 
tribution of  velocities,  we  may  employ  the  following  indirect  demon- 
stration given  by  Maxwell  in  his  paper  in  Phil.  Mag.  (4),  19,  1860, 
reprinted  in  his  Scientific  Papers,  Vol.  I.,  paper  XX.,  Prop.  V.,  pp. 
382-3. 

Consider  two  sets  of  molecules,  whose  velocities  we  may  represent 
by  c  and  c',  and  their  relative  velocity  by  r.  Let  the  components  be 
given  by  the  three  equations 

c*  =  u2  -f  v2  -f  w*,  • 


MOLECULES    HAVING    DIMENSIONS.  6 1 

suppose  the  molecules  to  have  all  the  same  speed  c, 
then  the  relative  speed  of  any  two  of  them  the  angle 
between  whose  directions  is  6  will  be  given  by  the 
equation 

r2  =  ?  4-  /—  2?  cos  0  =  2?(i  —  cos  0) 
or 


r  —  cV2(i  —  cos  6). 

Let  the  numbers  of  each  kind  in  unit  space  be  n  and  »'  and  the  prob- 
able speeds  be  a  and  j3.  Then  the  number  of  the  first  sort  having  the 
JSf-components  of  their  velocities  between  u  and  u  -f-  du  is,  according 
to  Maxwell's  law  (p.  22,  (6)), 


Similarly,  the  number  of  the  second  sort  having  components  between 
u  -\-  £  and  u  -f-  £  -j-  d%  is 


These  forms  hold  for  any  value  of  w,  and  the  number  of  pairs  of  mole- 
cules, one  of  each  sort,  having  the  relations  here  given  is  the  product 


. 

nnf      ~|  a«          02 
—  -  -       ua  p 


, 

du 


and  the  total  number  of  such  pairs  we  may  find  by  integrating  this  ex- 
pression for  all  values  of  u  from  —  co  to  +  oo  ,  giving 


where 


Now  put 


62  KINETIC    THEORY. 

Now  the  relative  number  of  molecules  whose  direc- 
tions make  an  angle  between  0  and  0  -f  d0  with  any 

sin  0  dO 
given  direction  has  been  found  to  be ,  hence 

we  can  get  the  average  value  of  r,  by  multiplying  its 

Then 


, 


The  number  above  is  then 

?*— -.*=**•'''*          * 

yV  y  a2  -\-  p2 
nn,          _  _!!_ 

=^e    a         * 


Now  this  is  the  expression  for  the  number  of  pairs  having  relative 
speeds,  the  .^-component  of  which  is  between  £  and  £  -f  ^?>  tne  whole 
number  of  possible  pairs  being  nn',  and  is  of  the  same  form  as  the  ex- 
pression for  the  number  of  molecules  having  the  Jf-components  of  their 
velocity  between  u  and  u  -\-  du,  which  is 


except  that  the  parameter  is  V a2  -f-  /32  instead  of  a.  This  demonstra- 
tion applies  equally  well  to  all  the  components  of  the  relative  speeds, 
and  hence  we  conclude  that  the  relative  speeds  of  the  two  sets  of  mole- 


MOLECULES   HAVING   DIMENSIONS.  63 

value  found  above  by  this  ratio,  and  integrating  from 
o  to  TT,  since  all  possible  directions  are  to  be  taken 
into  account.  This  gives  us 


OdO      -      *      I-COS0 


c  I     sin  -  sin  6  d0  =  AC  I    sin2  -  cos  -  ^  - 

Jo  2  Jo  2  2        2 


This  gives  us 

cules  follow  Maxwell's  distribution,  but  with  the  parameter  V  a2-\-  (32. 

Then  as 

2 

<•  =  —  —  a, 

1/7T 


1/7T 

and  if  the  two  systems  are  one  and  the  same  system  so  that 


The  same  result  is  obtained  by  assuming  that  the  one  molecule,  with 
speed  c,  is  moving  in  a  space  in  which  all  the  other  molecules  are  mov- 
ing with  the  same  speed  c,  but  in  planes  perpendicular  to  the  direction 
of  motion  of  the  first  molecule.  Then  in  the  equation 


r=c 2(1  — cos 
cos  •&  —  o,  and  hence 


64  KINETIC   THEORY. 


and  /_ 

(24) 

The  more  accurate  assumption  that  the  molecules 
have  not  all  the  same  speed,  but  have  Maxwell's  dis- 
tribution gives  us 

c         i 

=  =  ^7=  =-707, 
r       1/2 

a  result  which  is  only  slightly  different,  but  much 
more  difficult  to  deduce.  Using  this  value, 


We  do  not  yet  know  a-  or  n  and  consequently  cannot 
use  them  to  compute  /,  but  /  may  be  determined  by 
other  methods,  so  that  this  equation  may  later  help 
us  to  find  the  number  and  size  of  the  molecules. 

We  may  also  ask  the  relative  number  of  the  mole- 
cules which  travel  any  given  distance  x  between  two 
successive  impacts.  To  learn  this  we  find  the  proba- 
bility that  a  molecule  will  travel  the  distance  ;r,  before 
meeting  another  molecule.  If  we  call  this  y,  the 
probability  that  it  will  travel  the  distance  x  -\-  dx  is 

dy    ; 
y  +  dy=y  +  ~dxdx. 

Now  by  the  theory  of  probabilities,  the  probability 
that  it  will  travel  both  the  distance  x  and  the  distance 


MOLECULES    HAVING    DIMENSIONS.  65 

dx  is  the  product  of  the  separate  probabilities.  The 
first  we  have  called  y.  The  second  we  can  find  as 
follows  :  on  p.  56  we  found  that  the  probability  that 
a  molecule  in  traveling  the  distance  cdt  should  hit 

another  molecule  was 

nscdt 


Similarly  the  probability  that  in  traveling  the  distance 
dx  it  should  hit  another  molecule  is 

nsdx      dx 


v      r 

and  the  probability  that  it  will  not  have  a  collision  is 

dx 
"   /' 

Then  the  probability  that  it  will  succeed  in  traveling 
both  the  distance  x  and  the  distance  dx  without  a  col- 
lision is  the  product 

d*\         .  dy  , 


Hence 

dy_ 

dx 


dy          dx 

7=    '•/* 

log  y=  log  C—  -,, 


66  KINETIC    THEORY. 

The  fact  that  in  going  no  distance  at  all  a  molecule 
will  meet  with  no  collisions  makes  y  =  I  for  x  =  o  and 
hence  the  value  of  the  constant  is  C=  I  and 

_  X 

y  =  e   *. 

Corrections  in  the  value  of  /  will  not  affect  the  form 
of  this  function.  That  this  is  so  we  may  find  by 
employing  this  function  to  find  the  average  value  of 
the  distance  traveled  by  a  molecule  between  two  suc- 
cessive collisions.  The  probability  that  a  molecule 
will  travel  some  distance  between  x  and  x  -f  dx  is 
given  by  the  absolute  value  of  dyjdx  dx  which  is 


of  all  the  n  molecules  in  a  unit  volume 


would  travel  this  distance,  and  the  sum  of  the  lengths 
of  all  their  paths  would  be 

1lx   -is 
-j-e    l  dx, 

and  the  average  value  of  x  would  be 
C~nx  _! 

1  T'   dx 


'0 

After  dropping  the  common  factor  n,  it  is  evident  that 


MOLECULES    HAVING   DIMENSIONS.  67 

the  integral  in  the  denominator  must  have  the  value  i, 
since  it  is  the  probability  that  the  length  of  path  lies 
between  o  and  oo.  Integration  gives  the  same  result 
very  easily.  The  numerator  may  be  evaluated  by 
integration  by  parts, 


X*  x 
j 


dx 


-[-*""']> 


This  result  serves  rather  as  a  check  on  the  correctness 
of  the  form  deduced  for  y  than  as  any  addition  to  our 
knowledge. 

Pressure.  —  We  have  now  to  enquire  whether  our 
previous  deduction  of  the  intensity  of  the  pressure  of 
a  gas  still  holds  good  when  the  volume  occupied  by 
the  molecules  cannot  be  disregarded.  In  the  deduc- 
tion we  assumed  that  the  centers  of  the  molecules  went 
clear  up  to  the  walls  against  which  the  pressure  was 
exerted,  whereas  they  actually  never  came  nearer  than 
a  distance  a/2  from  them,  then  in  our  deduction  of 
the  pressure  if  we  disregarded  the  mutual  collisions  of 
the  molecules  we  should  still  have  to  replace  the  dis- 
tance h  between  the  parallel  walls  by  h  —  cr,  the  dis- 
tance a  molecule  would  pass  in  going  straight  from  one 
wall  to  the  other.  Now  on  the  average  each  molecule 
travels  a  distance  /,  and  that  in  a  direction  making  an 
angle  6  with  the  normal  to  the  plane  on  which  it  exerts 
its  pressure,  so  that  the  component  of  its  motion  per- 
pendicular to  the  plane  is  /  cos  6,  and  we  can  find  the 


68  KINETIC   THEORY. 

effect  of  the  impacts  by  taking  the  distance  between 
our  two  planes,  k  =  l  CQS  0  +  ^ 

Then  the  sum  of  all  the  impacts  of  a  single  molecule 
in  one  second  (p.  33,  (14)),  will  be 

me*  cos2  6   -  me1  cos2  6      mcz 

—  r  —       -  =  —  i  --  a~  =  —r  cos  0> 
h  —  o-  I  cos  6  I 

while  the  number  having  directions  between  9  and  0 
+  a»will  be  (p.  34,  (15)) 

nhs  sin  6  d6  =  n(l  cos  6  +  a)s  sin  0  d6, 
so  that  the  total  impulse  due  to  these  in  one  second 

is  the  product 

tttc 

n(l  cos  0  -f  a)s  sin  0  dO  -j-  cos  6 

-=  I  cos  0  +  o- 
=  nmrs  -    —,  —  -  cos  0  sin  0  </0 

and  the  pressure  required  to  equilibrate  these  impulses 
is  this  quantity  divided  by  s,  or  the  total  pressure  due 
to  the  molecules  moving  in  all  directions  is 


p/2  urn?  (I  cos  0  +  <r) 
p  =  —  --  -,  -  -  cos  0  sin  0  dv 

Jo  * 

nm?  f"  P2  cos2  9M1Hi  +  -A  "cos  0  sin  0  dO\ 

-  f  a  ~}nl2 

=  nmc*    -  J  cos3  ^  -  \~t  cos2  (9 


MOLECULES   HAVING   DIMENSIONS.  69 

which  may  be  written 


or  letting  nv  —  N  and  considering  that  b  is  small  in 
comparison  with  v 

(26)  p(v-b)  =  $Nm?. 

This  is  evidently  an  extension  of  the  equation  for  ideal 
gases  and  can  also  be  written 

(27)  f(v-b)  =  RT 

and  is  of  interest  because  it  expresses  almost  exactly 
the  behavior  of  hydrogen. 

We  have  in  obtaining  this  form  made  the  substitution 

*_3- 

V~*   I 

which  gives  for 


-_ 

47T770-2' 

b  —  2'jrvna3'  =  N-  27r<r3. 
The  total  volume  of  all  the  molecules  is 


If  we  had  used  the  value 


1/27T7/0-2 

we  should  have  found 


3T/27T7Z0-3          37JV20-3 


70  KINETIC   THEORY. 


1/2 

and  the  total  volume  of  all  the  molecules 
=  ^  £  =  .0786^. 

Or  in  other  words,  b  is  12,  or  12.7  times  the  volume 
of  the  molecules.  This  correction  b  is  much  too 
large,  for  we  are  interested  particularly  in  the  internal 
pressure,  rather  than  that  upon  a  wall,  and  the  value 
of  the  mean  free  path  was  deduced  for  motions  in  the 
interior  of  the  gas.  The  collisions  may  be  of  every 
sort  from  exactly  central  to  exactly  grazing,  so  that 
-the  correction  for  an  end  of  the  path  instead  of  being 
a  1  2  may  have  any  value  from  o  to  cr/2,  on  the  average 
(7/4,  and  this  correction  will  be  in  the  direction  of  /, 
hence  the  pressure  will  be 


C 
=   I 


Sin 


gvng 


instead  of  |  <rjl  ;  hence  b  is  not  1  2  but  4  times  the 
volume  of  the  molecules.  This  is  the  value  found  by 
van  der  Waals,  while  Clausius  and  O.  E.  Meyer  find 
the  ratio  41/2.  While  the  ratio  is  then  slightly  in 
doubt,  it  is  still  evident  that  b  is  some  small  multiple 


MOLECULES    HAVING   DIMENSIONS.  /I 

of  the  volume  of  the  molecules,  and  of  the  same  order 
of  magnitude. 

Specific  Heats,  —  In  the  case   of  molecules  whose 
size  cannot  be  entirely  disregarded  we  can  no  longer 
assume  that  the  only  motion  of  the  molecules  is  their 
motion  of  translation,  or  that  their  energy  is  all  energy 
of  translation.    In  the  case  of  ordinary  bodies  of  consid- 
erable dimensions  we  observe  a  continual  tendency  for 
the  motion  of  the  body  to  degenerate,  in  consequence 
of  friction,  more  and  more  into  motion  of  the  smaller 
parts,  and  vibratory  motions  of  greater  and  greater 
complexity  and  smallness,  till  we  say  their  energy  is 
dissipated  in  heat.     In  almost  all  terrestrial  motions 
this  tendency  is  quite  marked,  and  the  degradation  of 
other  forms  of  energy  into  heat  is  very  rapid,  conse- 
quently many  have  found  difficulty  in  securing  a  satis- 
factory  conception    of  the    kinetic    theory.       It   has 
seemed  to  them  that  with  every  collision  of  the  mole- 
cules, just  as  with  every  collision  between  extended 
bodies,  there  must  be  a  degradation  of  a  large  part  of 
their  energy  into  energy  of  the  motion   of  smaller 
parts,  so  that  there  would  be  no  state  of  real  equilib- 
rium.    There  is  this  important  difference,  however, 
between  the  two  cases  which  are  thus  compared,  even 
if  we  consider  our  molecules  to  have  dimensions,  the 
smaller  parts  are  not  indefinitely  smaller  than  the  mole- 
cules, consequently  at  every  collision  there  will  be  a 
redistribution  of  the  energy  between  the  different  pos- 
sible modes  of  motion  of  the  molecules,  the  vibratory 
or  rotary  motion  of  the  molecule  being  sometimes  in 


72  KINETIC   THEORY. 

such  a  phase  at  the  time  of  the  collision  as  to  result 
in  an  increase  in  the  energy  of  translation,  sometimes, 
in  such  a  phase  as  to  take  up  more  of  the  energy  in 
the  internal  motions  of  the  molecule ;  a  similar  phe- 
nomenon is  observed  when  an  ivory  ball  suspended  by 
a  thread  rests  in  contact  with  the  end  of  a  metal  rod. 
If  longitudinal  vibrations  are  excited  in  the  rod  by 
rubbing  it,  the  ivory  ball  is  thrown  violently  away 
from  its  position  and  on  its  return  may  have  its  mo- 
tion almost  checked,  or  it  may  rebound  with  increased 
violence  according  to  the  phase  of  the  vibration  of  the 
rod. 

It  appears  that  the  vibrational  motions  do  not  need 
to  be  considered  in  the  case  of  perfectly  elastic  solids, 
such  as  we  are  considering,  for  the  forces  of  deformation 
on  collision  expend  themselves  completely  in  effecting 
the  rebound,  or  in  other  words,  the  coefficient  of  res- 
titution of  the  molecules  is  unity.  This  view  is  upheld 
by  Lord  Kelvin,  a  former  opponent,  who  says  : * 

"  /  now  see  that  the  average  tendency  of  collisions 
between  elastic,  vibrating  solids  must  be  to  diminish  the 
vibrational  energy,  provided  the  total  energy  per  indi- 
vidual solid  is  less  than  a  limit  depending  on  the  shape 
or  shapes  of  the  solids  ;  and  hence,  as  nothing  is  lost  of 
the  whole  energy,  conversion  of  all  but  an  infinitesimal 
portion  into  translational  and  rotational  energy  must  be 
the  ultimate  result" 

In  a  state  of  equilibrium,  then,  there  will  be  for  any 
temperature  and  for  a  definite  kind  of  molecule,  a 

1  "  Popular  Lectures  and  Addresses,"  Vol.  I.,  p.  464. 


MOLECULES    HAVING   DIMENSIONS.  73 

definite  ratio  between  the  average  values  of  the  kinetic 
energy  of  translation  and  of  the  internal  motion  of 
the  molecules.  Inasmuch  as  this  distribution  can  only 
depend  upon  the  effects  of  the  collisions,  and  the  phe- 
nomena of  a  collision  can  depend  only  on  the  velocities 
of  the  molecules,  and  not  on  the  distances  they  travel 
between  collisions,  this  ratio  will  depend  only  on  the 
temperature  of  the  gas,  and  not  upon  its  density,  that 
is,  it  will  not  be  changed  if  the  gas  is  expanded  at 
constant  temperature.  Let  us  see  what  effect  this  will 
have  upon  our  conceptions  of  the  specific  heats  of  the 
gas.  The  amount  of  heat  required  to  heat  the  gas 
from  the  temperature  7^  to  T2,  representing  the  other 
corresponding  properties  by  the  indices  I  and  2,  will 
consist  of  two  parts,  one,  which  we  will  call  K,  which 
increases  the  kinetic  energy  of  translation  of  the  mole- 
cules from  \Nrnc*  to  \Nrnc  •*,  the  other  k,  which  in- 
creases the  internal  energy  of  the  molecules  by  the 
amount  corresponding  to  the  rise  of  temperature.  Then 

K+k=  \Nm(ct  -  ^)  +  k  =  CV(T2  -  T;). 

If  the  gas  be  heated  over  the  same  range  of  temper- 
ature but  kept  at  constant  pressure  instead  of  at  con- 
stant volume,  besides  the  amount  of  heat  just  speci- 
fied there  would  also  be  as  in  the  case  of  an  ideal  gas 
the  amount  required  to  do  the  work  of  expansion  from 
the  volume  vl  to  the  volume  vv  which  we  found  to  be 


Then  the  total  amount  of  heat  would  be 


74  KINETIC   THEORY. 

and  the  ratio  oi  the  two  specific  heats  would  be 
C  - 


f 


or  if  we  call  the  sum  K  -f  k  =  H, 

-c'-i  +  t-. 

V 

from  which  we  may  deduce  the  relation 
K 


In  the  above  expression  H  is  the  energy  required  to 
raise  the  temperature  of  the  gas  from  the  tempera- 
ture 7^  to  Tv  and  K  is  the  part  of  this  energy  which 
increases  the  kinetic  energy  of  translation  of  the  mole- 
cules. If  the  ratio  of  these  two  is  independent  of  the 
temperature,  then  KjH'is  the  ratio  of  the  energy  of 
translation  of  the  molecule  to  the  total  energy.  In 
any  case 

K 
°<ff<*. 

and  hence 

K  £  <  if- 

Ct> 

The  above  treatment  is  due  to  Clausius.  In  the  case 
of  an  ideal  gas,  which  is  also  monatomic,  KjH  may 
approach  its  upper  limit,  unity,  giving  us  as  before  the 
limiting  value 


MOLECULES   HAVING   DIMENSIONS.  75 

We  may  obtain  a  still  more  definite  evaluation  of  the 
ratio  of  the  specified  heats  by  means  of  a  theorem  due 
to  Boltzmann.1  This  theorem  is  founded  upon  a  gen- 
eralized conception  associated  with  the  phrase  "  de- 
grees of  freedom."  The  number  of  degrees  of  freedom 
of  an  object  is  the  number  of  facts  which  must  be 
specified  in  order  to  describe  completely  its  state,  or 
in  the  more  restricted  case  with  which  we  are  con- 
cerned, its  position.  For  instance,  the  position  of  a 
point  is  determined  by  three  coordinates  ;  these  may 
be  the  three  rectangular  coordinates,  x,  y,  z ;  or  the 
coordinates  of  the  polar  or  geographical  system,  ry  9y 
<j) ;  or  any  three  independent  coordinates  of  any  suit- 
able system,  but  three  is  the  smallest  number  of 
coordinates  which  can  define  the  position  of  a  point. 
Similarly  two  points  are  completely  specified  by  six 
coordinates,  but  if  there  is  some  definite  relation  be- 
tween these  two  points,  as  for  instance  an  equation 
stating  their  distance  apart,  the  number  of  coordinates 
necessary  for  a  complete  description  is  reduced  by  one ; 
for  instance,  we  may  choose  the  three  coordinates  of 
one  of  the  points  and  the  two  angular  coordinates  6 
and  <f>  which  will  give  the  direction  of  the  line  joining 
the  two  points,  and  these  five  coordinates  will,  with 
the  knowledge  of  the  distance  of  the  points,  determine 
the  position  of  the  system,  which  is  then  said  to  have 
five  degrees  of  freedom.  In  general,  the  number  of 
degrees  of  freedom  is  equal  to  the  number  of  coordi- 

1  Boltzmann,  "  Gastheorie,"  II. ,  pp.  125-130. 


76  KINETIC   THEORY. 

nates  required  to  define  the  positions  of  the  elements 
of  the  system,  decreased  by  the  number  of  indepen- 
dent relations  existing  between  these  elements.  Hence 
if  we  call  the  number  of  atoms  in  the  molecule  of  our 
gas  ;/,  the  number  of  degrees  of  freedom  cannot  ex- 
ceed 3# ;  for  a  monatomic  gas,  with  n=  I,  it  will  be 
3  ;  for  a  diatomic  gas,  having  the  two  atoms  at  a  fixed 
distance  from  each  other  it  will  be  5,  as  we  have  just 
shown.  For  three  atoms  we  may  have,  according  to 
the  arrangement,  either  a  central  atom  with  the  two 
others  swinging  from  it,  or  the  three  at  the  apices  of 
a  triangle.  The  number  of  degrees  of  freedom  will 
then  be  in  the  first  case 

3/2  —  2  =  9-2  =  7 
or 

3«  -  3  =  9  -  3  =  6. 

And  for  more  complex  molecules  the  formulae  may 
be  still  more  complex,  and  the  number  of  degrees  of 
freedom  much  greater.  Now  Boltzmann's  theorem  is 
that  as  a  result  of  all  the  impacts  between  the  mole- 
cules, their  kinetic  energy  tends,  on  the  average,  to  be 
equally  distributed  among  the  motions  corresponding 
to  the  different  freedoms.1  This  is  best  explained  by 
applying  it  to  the  types  of  molecules  we  have  just 
been  discussing.  The  number  of  degrees  of  freedom 
associated  with  pure  translation  is  evidently  3,  the 
number  of  coordinates  in  space.  Then  for  a  mona- 

1See  Rayleigh,  Phil.  Mag.   (5),  49,  pp.  98-118,  190x5.     Kelvin, 
Phil.  Mag.  (6),  2,  pp.  1-40,  1901. 


MOLECULES    HAVING   DIMENSIONS.  77 

tomic  gas,  having  only  three  degrees  of  freedom  for  its 
molecule 

K 


for  a  diatomic  gas,  whose  molecule  has  5  degrees  of 
freedom, 


for  a  triatomic  gas,  according  to  its  arrangement 

K 
ff==^> 

or 

._  3 i 

Tf  ~  *  ~  *  • 

We  may  substitute  these  values  of  KjH  in  the  formula 

C,  * H* 

obtaining  the  values 

n=  I  -£  =  1.66, 

2  1.4, 

3  (linked)  1.28, 
3  (triangular)             1-33- 

We  have  already  mentioned  the  fact  that  the  value 
of  this  ratio  for  mercury  vapor,  which  is  believed  to 
be  monatomic,  is  found  to  be  1.666.  The  diatomic 


7$  KINETIC   THEORY. 

gases,  oxygen,  nitrogen,  air  (a  mixture  of  the  two 
preceding),  hydrogen,  and  some  others  give  values 
which  average,  for  different  experimenters,  about 
1.405.  The  haloid  elements,  chlorine,  bromine,  iodine, 
appear  to  give  values  in  the  neighborhood  of  1.3, 
but  these  values  are  anomalous,  and  may  be  due  to 
approaching  dissociation.  Of  the  triatomic  gases,  the 
ratio  for  carbon  dioxide,  CO2  is  given  by  various  ob- 
servers all  the  way  from  1,265  to  1.311  ;  for  nitrous 
oxide,  N2O,  from  1.27  to  1.31 1  ;  for  sulphurous  oxide, 
SO2,  from  1.248  to  1.262,  and  for  hydrogen  sulphide, 
H2S,  from  1.258  to  1.276.  The  agreement  of  these 
numbers  with  those  suggested  by  the  theoretical  dis- 
cussion is  close  enough  to  give  the  discussion  great 
interest,  and  is  nearly  as  close  as  the  agreement  be- 
tween the  results  of  different  observers.  There  still 
remain,  however,  differences  of  sufficient  magnitude 
so  that  they  must  be  recognized ;  these  we  shall  con- 
sider in  a  later  chapter. 


CHAPTER   IV. 
TRANSPORT   PROBLEMS. 

OUR  discussion  of  gaseous  phenomena  has  so  far 
been  confined  to  cases  in  which,  whatever  the  motions 
of  the  molecules  individually,  the  gas  as  a  whole  was 
at  rest  and  in  equilibrium.  This  state  of  equilibrium 
has  been  one  of  the  fundamental  assumptions  upon 
which  the  treatment  has  rested.  We  have  now  to  take 
up  a  class  of  problems  in  which  we  observe  not  equi- 
librium, but  simply  a  steady  state,  not  simply  of  the 
molecular  motions,  but  also  of  the  gas  itself.  Exam- 
ples of  such  problems  are  the  conduction  of  electricity 
and  heat  by  gases ;  the  flow  of  gases  through  tubes, 
and  other  phenomena  of  gaseous  viscosity,  and  the 
phenomena  of  diffusion. 

Conduction  of  Electricity. —  Perhaps  the  simplest  of 
these  problems  is  that  of  the  conduction  of  electricity 

under   the    purely  hypo- 

g 

thetical    assumption    that    j 

the  molecules  are  perfect  j 

conductors  of  electricity.     


Suppose  two  planes,  which 

we  may  designate  by  the  subscripts  i  and  2,  to  be  at 
a  definite  distance  d  apart,  and  to  be  kept  charged 
at  the  potentials  V^  and  V2  respectively.  If  we  make 
these  planes  perfectly  conducting  plates,  each  mole- 

79 


80  KINETIC   THEORY. 

cule  coming  in  contact  with  one  of  the  planes  will  re- 
ceive from  it  such  a  charge  as  will  bring  it  just  to  the 
potential  of  that  plane.  Call  these  charges  Gl  and  G2. 
If  the  molecules  are  perfect  spheres  of  diameter  cry  the 
electrical  capacity  of  each  molecule  will  be  <r/2,  and 
hence 


For  the  sake  of  definiteness  we  may  consider  the 
potential  V2  higher  than  V^  and  the  charge  G2  greater 
algebraically  than  Gr  Then  in  any  other  plane  inter- 
mediate  between  the  planes  I  and  2  the  potential  will 
be  intermediate  between  Vl  and  Vv  and  the  average 
charges  of  the  molecules,  whether  by  the  equalizing 
effect  of  interchange  of  charges  upon  collision  or  by 
the  mixture  of  molecules  coming  from  the  two  oppo- 
site regions,  will  have  a  perfectly  definite  value  inter- 
mediate between  Gl  and  G2.  If  we  suppose  that  suf- 
ficient time  has  elapsed  for  the  establishment  of  a 
steady  state  of  conduction,  and  that  the  plates  are  of 
large  area  compared  with  the  distance  between  them, 
the  potential  and  the  average  charge  of  the  molecules 
can  be  expressed  as  linear  functions  of  the  distance  of 
the  plane  we  are  considering  from  the  two  planes  of 
reference.  If  we  take  the  Z-axis  perpendicular  to 
these  planes,  measuring  z  positively  from  the  plane  I 
toward  the  plane  2,  then  the  potential  and  average 
molecular  charges  at  any  point  are  given  by  the 

expressions 

V  —  V 


TRANSPORT   PROBLEMS.  8  1 


which  reduce  to  FJ  and  Gl  respectively  at  the  plane 
I,  where  z  =  o,  and  V.2  and  G2  at  the  plane  2,  where 
z  =  d.  The  molecules  arriving  at  any  plane  Pt  par- 
allel to  the  planes  I  and  2,  will  each  bring  with  them, 
on  the  average,  the  charges  corresponding  to  the 
plane  in  which  they  experienced  their  last  collisions. 
Now  we  have  no  means  of  knowing  just  what  distance 
has  been  traveled  by  each  molecule,  and  still  less  do 
we  know  what  proportion  of  its  total  path  between 
the  last  previous  and  the  next  following  collisions  has 
been  passed  before  reaching  this  plane.  But  we  have 
no  reason  for  considering  the  probability  that  a  given 
molecule  shall  strike  or  pass  through  a  given  portion 
of  this  plane  any  different  from  that  for  any  other 
equal  area  ;  hence  we  must  assume  that  the  molecules 
passing  through  this  plane  have  on  the  average  trav- 
eled a  distance  equal  to  the  mean  free  path  of  the 
molecule  since  their  last  previous  collisions. 

Then  any  molecule  reaching  a  plane  whose  distance 
from  the  plane  I  is  zt  having  traveled  a  distance  / 
since  its  last  collision  in  a  direction  making  an  angle 
d-  with  the  normal  to  the  plane,  will  have  come  from  a 
plane  higher,  or  lower  than  this  plane  by  a  distance 
/  cos  #,  that  is,  from  a  plane  whose  distance  from  the 
plane  I  is 

z  -f  /  cos  $  or  z  —  /  cos  ft, 

and  such  molecules  will  bring  with  them  on  the  aver- 

6 

,. 


82  KINETIC   THEORY. 

age,  the  charges  corresponding  to  these  planes,  that  is, 


and 

G!  +  &=£(*-  1  coat), 

If  we  assume  that  all  the  molecules  have  the  same 
speed,  r,  the  molecules  coming  from  a  direction  making 
the  angle  $  with  the  axis  of  Z  will  have  a  component 
Velocity  in  the  direction  Z. 

w  =  c  cos  $. 

The  number  of  such  molecules  in  unit  volume  will  be 

(P-  34,  (IS)), 

n  .     _   7. 
~  sm  &  dd-  : 


and  the  number  of  these  which  will  pass  through  unit 
area  normal  to  the  axis  ofZ'm  one  second  is 

w  .  -  sin  &  d$  —  ±nc  sin  &  cos  &  d$ 

Integrating  this  from  o  to  7r/2  we  get  as  the  total 
number  of  molecules  passing  through  unit  area  from 
one  side  in  one  second 


.  r«ft 

\nc  I      si 

Jo 


sin  $  cos  &  dd-  =  \nc  [J  sin2  #]  ^   =  \n~cy 


a  result  which  may  be  found  either  by  more  complex 
methods,  or  by  taking  the  average  value  of  cos  $,  J 
(P-  59). 


TRANSPORT    PROBLEMS.  83 

Each  of  these  molecules,  having  come  a  distance  / 
will  have  come  from  a  layer  whose  normal  distance  is 
/  cos  $,  and  the  average  of  these  values  will  be 

Jir/2                                 _                                                               /»7T/2 
/  cos  #  -  \nc  sin  #  cos  $  d&          I      sin  &  cos2  &  dd 
7  «/0 


X7T/2  .   _  /%»/a 

J  w  sin  i?  cos  #  a$  si 

Jo 


sn      cos 


- 
[Isin'tfU"        'p 

Using  these  values,  the  average  charges  brought  with 
them  by  the  molecules  passing  through  the  plane  z 
are 


and  the  total  quantities  carried  by  the  molecules  pass- 
ing downward  and  upward  respectively  through  the 
planes  will  be 


The  resultant  current  of  electricity  passing  through 
unit  area  of  the  plane  will  be  given  by  the  difference 
of  these  quantities,  which  is, 


84  KINETIC   THEORY. 

or,  if  the  molecules  are  perfect  spheres, 


-, 
2  =  *  ncl<7 


in  which  (  V^  —  V^)  /  d,  which  may  also  be  written 
d  VI  dZ,  is  the  potential  gradient,  and  \  ncla-  the  specific 
conductivity.  If  we  give  to  /  its  value  (p.  64,  (24)) 
3/47r;zcr2,  computed  on  the  assumption  that  all  the  mole- 
cules have  the  same  speed,  we  have  for  the  conductivity, 


an  expression  involving  only  c  and  o\ 

This  deduction  can  make  no  claim  to  numerical 
accuracy,  on  account  of  the  arbitraiy  assumptions  and 
approximations  made.  Irregularities  in  the  shape  of 
the  molecules  might  change  the  electrical  capacity 
slightly  but  would  not  seriously  affect  the  numerical 
results.  Of  more  importance  would  be  the  influence 
of  the  mutual  forces  exerted  between  the  charged  mole- 
cules upon  their  motions,  which  we  have  neglected  en- 
tirely. Further  we  have  made  our  deductions  only 
approximate  by  assuming  that  all  the  molecules  have 
the  same  speed,  and  by  averaging  separately  the  effects 
of  the  inclination  of  the  paths  of  the  molecules  upon 
the  charges  carried  by  the  molecules,  and  upon  the 
numbers  of  molecules  passing  through  the  plane. 
Boltzmann  x  using  more  exact  methods  finds  the  con- 

ductivity to  be  k 

~ 

K'Gastheorie,"  I.,  p.  80. 


TRANSPORT   PROBLEMS  85 

where  k  is  a  definite  integral  whose  value  he  finds  to 
be  .35027,  thus  making  the  numerical  coefficient 
.17514,  while  the  approximate  method  we  have  em- 
ployed makes  it  \  =  .1667 

These  values  which  we  have  obtained  are  based 
upon  the  fundamental  assumption  that  the  molecules 
are  perfect  conductors,  an  assumption  which  is  con- 
fessedly not  in  accordance  with  the  facts  as  we  know 
them,  and  hence  the  formulae  deduced  cannot  be  em- 
ployed to  compute,  from  observation  upon  the  con- 
ductivity, numerical  values  of  <7,  the  diameter  of  the 
molecules. 

Viscosity  of  Gases. — This  problem  differs  from  the 
one  just  treated  in  that  the  difference  between  the 
molecules  in  different  regions  is  a  difference  in  veloc- 
ities, and  not  a  simple  difference  in  some  extraneous 
quality,  which  does  not  affect  the  motions  of  the  mole- 
cules. We  shall  suppose  that  the  gas  we  are  consider- 
ing is  moving  as  a  whole  in  one  direction,  but  different 
portions  with  different  speeds.  For  convenience  we 
may  take  the  direction  of  the  motion  as  the  direction 
of  the  Jf-axis  ;  for  the  sake  of  definiteness  and  sim- 
plicity we  shall  assume  that  all  the  gas  in  any  plane 
perpendicular  to  the  Z-axis  has  the  same  general 
motion,  and  since  all  motion  is  relative,  we  will  take 
the  plane  I  (Fig.  7)  as  the  plane  of  no  motion.  Call 
the  velocity  of  the  plane  2  in  its  own  plane  in  the 
direction  of  X,  VQ.  Then  if  the  distance  of  any  point 
perpendicularly  frcm  the  plane  I  toward  the  plane  2 
be  called  z,  and  if  the  motions  of  the  gas  are  steady, 


86  KINETIC   THEORY. 

we  may  expect  that  the  general  motion  of  the  gas  at 
any  intermediate  point  will  be  a  linear  function  of  its 
distance  from  the  plane  i,  or 

dV 


where,  if  d  be  the  distance  between  the  planes  i  and  2, 


This  gives  us  an  expression  for  the  motion  of  the  gas 
as  a  whole.  The  motions  of  the  individual  molecules 
will  be  the  resultant  or  sum  of  the  motions  they  would 
have  if  the  gas  were  at  rest,  and  the  motion  of  that 
part  of  the  gas  in  which  they  happen  to  be,  so  that 
the  three  components  of  their  velocities  will  be  not 
«,  v,  w,  but  u  -f  V,  v,  w,  where  u,  v,  w,  and  their  re- 
sultant c  represent  the  ordinary  velocity  of  a  molecule 
when  the  gas  as  a  whole  is  at  rest.  Now  the  velocity 
Vt  even  if  it  corresponded  to  a  very  violent  motion  of 
the  gas,  would  still  be  very  small  as  compared  with  c, 
the  average  speed  of  the  individual  molecules.  A 
wind  of  fifty  miles  an  hour  is  a  destructive  gale,  and 
one  of  a  hundred  miles  an  hour  is  capable  of  destroy- 
ing everything  that  comes  in  its  path,  while  c,  for  air, 
is  of  the  order  of  a  thousand  miles  an  hour.  Conse- 
quently we  can  without  serious  error  regard  the  mole- 
cules in  one  layer  as  differing  from  the  molecules  in 
another  layer  only  in  the  possession  of  different  quan- 
tities of  directed  momenta,  the  direction  of  these  being 
the  same  for  all  layers,  namely  the  direction  of  the 


TRANSPORT   PROBLEMS.  87 

motion  of  the  gas,  parallel  to  the  X-axis,  but  the 
amount  varying  from  layer  to  layer  with  the  value  of  z. 
From  the  dynamical  standpoint  the  effect  of  these 
differences  of  speed  upon  the  two  planes  is  a  force 
dragging  or  holding  back  the  plane  2  and  a  force 
tending  to  pull  along  the  plane  I.  Experiment  and 
theory  both  indicate  that  this  force,  which  seems  to 
be  due  to  an  internal  friction  of  the  gas  is  equal  in 
amount  for  the  two  planes,  is  proportional  to  their 
area,  to  their  difference  of  speed,  and  inversely  pro- 
portional to  the  distance  between  the  planes.  Then 
the  force  acting  on  a  unit  area  of  either  plane  may  be 
written 


where  rj,  the  factor  of  proportionality,  may  depend 
upon  the  nature  of  the  gas,  and  is  called  its  Coefficient  \ 
of  Viscosity. 

From  the  molecular  standpoint  molecules  striking 
the  plane  2,  will  have  on  leaving  it,  by  reason  of  their 
friction  with  it,  or  momentary  entanglement  with  it,  on 
the  whole  an  excess  of  velocity  VQ  in  the  direction  of 
the  motion  of  the  plane  and  receive  from  it  whatever 
addition  of  momentum  is  necessary  to  bring  them  up 
to  this  velocity.  Similarly  molecules  striking  the 
plane  I  will  give  up  to  it  all  their  excess  of  momentum, 
and  leave  it  with  velocities  such  as  they  would  have 
if  the  gas  as  a  whole  were  at  rest.  This  transfer  of 
momentum  from  the  plane  2  to  the  molecules  of  the 
gas,  and  from  these  to  the  plane  I  constitutes  the 


88  KINETIC   THEORY. 

mechanism  of  the  forces  observed.  The  force  must 
be  numerically  equal  to  the  amount  of  momentum 
transferred  in  one  second. 

This  transfer  will  take  place  uniformly,  on  the 
whole,  throughout  the  body  of  the  gas,  hence  we  have 
only  to  find  the  excess  of  this  directed  momentum 
carried  in  one  direction  by  the  molecules  of  the  gas 
through  any  plane  parallel  to  the  planes  I  and  2  over 
that  carried  in  the  other  direction.  The  method  of 
treatment  is  entirely  analogous  to  that  of  the  last 
problem.  The  molecules  coming  from  above  will 
come  from  an  average  height  J  /  above  the  plane  we 
are  considering,  and  will  carry  with  them  downward 
through  the  plane  an  average  directed  momentum 


while  those  from  below  will  bring  an  average  of 
y 


The  number  of  molecules  passing  thround  unit  area 
of  the  plane  in  one  second  will  be,  as  before, 

\ncy 

so  that  the  total  amount  of  directed  momentum  car- 
ried by  the  molecules  passing  downward  and  upward 
respectively  through  unit  area  in  one  second  is 


TRANSPORT   PROBLEMS.  89 

and  the  excess,  which  is  equivalent  to  the  force  exerted 
on  the  plane,  is 


whence 

(28)  ?; 

an  expression  whose  form  reminds  us  of  that  for  the 

pressure, 

(!)  /  =  \nrnf, 

a  length  /  replacing  in  it  a  velocity  c.  It  will  be  noted 
however  that  the  dimensions  are  entirely  different.  The 
more  exact  value  found  by  Boltzmann1  is 

knmlc, 

where  k  has  the  value  .350271,  not  very  different  from 
the  coefficient  we  have  found,  ^.  As  the  viscosity  of 
a  gas  can  be  determined  experimentally,  it  is  evident 
that  the  formula 


or  the  more  exact  form  given  by  Boltzmann  may  be 
employed  to  compute  the  value  of  /.  Some  of  the 
data  and  the  results  thus  obtained  will  be  given  in  a 
later  chapter. 

Substituting  in  the  formula  for  /  its  value  (p.  64, 

(24)), 


47T7ZCT2' 

-       3           me 
rj  =  \nrnc 2  = ; 

47T7Z0-2         47TO-' 


i«Gastheorie,"  I.,  p.  81. 


90  KINETIC   THEORY. 

This  value  would  by  more  accurate  methods  of  deduc- 
tion only  have  its  numerical  constants  slightly  changed, 
its  general  form  would  remain  the  same.  It  is  evident 
that  this  formula  cannot  be  employed  to  determine  the 
value  of  either  m  or  <r  unless  we  already  know  the  value 
of  one  of  them. 

The  last  formula  shows  that  the  coefficient  of  vis- 
cosity may  depend  upon  the  mass  of  the  molecules, 
upon  their  size,  since  o-2  is  involved,  and  upon  the  tem- 
perature, being  proportional  to  the  square  root  of  the 
latter ;  it  will  be  independent  of  the  pressure  or  den- 
sity of  the  gas,  though  not  of  its  kind  since  n  does  not 
appear.  We  may  express  the  viscosity  directly  as  a 
function  of  the  temperature  by  eliminating  the  velocity 
by  the  equations 

pv  =  \Nm?  =  RT, 
(28)  rj  =  \nmlc, 


These  give  us 


Sc2  \2RT 

c  =  A! =  2  <\|— ^, 

37T  \  irNm 


rj  =  f  nml\(  — -— » 

\irNm 

which  becomes  if,  we  use  the  other  value  of  /, 


TRANSPORT   PROBLEMS.  9 1 


(25) 


1/27T7ZCT2 

nm 


In  which  Nm  is  the  quantity  of  gas  taken  as  the 
standard  amount  in  computing  the  value  of  R,  for 
instance  one  gram  molecule,  so  that  RjNm  is  a  con- 
stant whose  value  depends  upon  the  particular  gas 
under  consideration,  while  RjN  in  the  last  expression 
is  independent  of  the  kind  of  gas.  This  form  shows 
even  more  strikingly  than  the  other  the  fact  that  the 
viscosity  of  the  gas  depends  only  upon  the  kind  of 
the  gas  and  its  temperature. 

Experiments  upon  the  viscosity  of  gases  have  shown 
that  the  viscosity  increases  with  the  temperature,  but 
is  not  exactly  proportional  to  the  square  root  of  the 
absolute  temperature.  Attempts  have  been  made, 
but  with  only  partial  success,  to  devise  formulae  which 
shall  be  able  to  express  the  dependence  of  the  vis- 
cosity upon  the  temperature.  Two  causes  for  varia- 
tion from  this  simple  formula  are  suggested,  the  mu- 
tual attractions  of  the  molecules,  and  change  of  aggre- 
gation, particularly  dissociation,  causes  whose  general 
effects  are  to  be  discussed  in  later  chapters. 


92  KINETIC   THEORY. 

We  have  seen  that  according  to  the  formula  devel- 
oped the  viscosity  of  a  gas  should  be  independent  of 
the  pressure  or  density.  Experiments  by  many  able 
investigators  have  shown  that  this  is  true  for  a  wide 
range  of  pressures,  but  below  1/60  of  an  atmos- 
phere,1 and  for  some  gases  above  30  atmospheres2 
there  is  some  variation  with  the  pressure.  It  is  in- 
deed to  be  expected  that  this  formula  would  fail  for 
extreme  cases,  since  it  was  developed  under  the  as- 
sumptions that  the  straight  portions  of  the  paths  of 
the  molecules  were  veiy  long  as  compared  with  the 
curved  portions  associated  with  their  mutual  impacts, 
which  is  less  nearly  justified  at  extremely  high  pres- 
sure, and  that  the  distances  between  the  solid  surfaces 
on  which  the  drag  is  exerted  are  large  as  compared 
with  the  mean  free  path,  an  assumption  whose  validity 
fails  when  the  pressure  becomes  exceedingly  small. 
The  discussion  of  this  last  case  will  be  taken  up  in 
connection  with  the  next  topic. 

Conduction  of  Heat.  —  The  statement  of  this  prob- 
lem is  entirely  similar  to  that  of  the  two  preceding 
problems.  The  gas  at  the  plane  I  is  at  the  tempera- 
ture Jj,  that  is,  it  has  a  mean  value  of  the  square  of 
the  velocities  of  the  molecules  c*,  and  the  correspond- 
ing values  at  the  plane  2  are  T2  and  c22.  The  temper- 
ature gradient  will  be 

T  —  T 

22  2\ 

d 

1  Kundt  &  Warburg,  Pogg.  Ann.,  1875,  CLV.,  pp.  337,  525. 

2  Warburg  &  Babo,   Wied.  Ann.,  1882,  XVII.,  p.  390. 


TRANSPORT   PROBLEMS.  93 

and  the   corresponding  rate   of  change  in   the  mean 
square  of  the  velocities  will  be 


so  that  the  mean  square  of  the  velocities  of  the  mole- 
cules in  any  plane  whose  distance  from  the  plane  I 
measured  toward  the  plane  2  is  z  will  be 


We  shall  assume  that  the  differences  of  temperatures 
and  density  are  small,  and  that  the  number  of  mole- 
cules passing  through  unit  area  in  each  direction  in 
unit  time,  which  must  be  the  same,  is  represented  by 
the  expression  previously  deduced,  namely 

\nc,   . 

and  that  the  molecules  come  from  the  same  average 
vertical  distance  J  /  above  or  below  this  plane,  then 
remembering  that  the  energy  of  a  molecule  is  given 
by  the  expression  J  mc?^  the  molecules  passing  down- 
ward through  unit  area  of  the  plane  in  one  second  will 
carry  with  them  the  energy 


and  similarly  the  amount  carried  by  those  passing  up- 
ward will  be 


94  KINETIC    THEORY. 


while    the    excess    of  the  amount  of  energy  carried 
downward  over  that  carried  upward  will  be 


_ 
\nrnc 


Since  the  heat  energy  of  the  standard  amount  of  gas, 
which  we  have  represented  by  ^Nmc2,  may  be  even 
more  accurately  represented  by  CVT,  we  may  substi- 
tute the  latter  expression  for  the  former,  giving  us 


in  which 


is  the  coefficient  of  thermal  conductivity.  The  ratio 
njN  is  defined  by  the  equation  N=  nv  as  the  recip- 
rocal of  v,  the  volume  occupied  by  the  standard 
amount  of  our  gas,  hence  the  coefficient  reduces  to 
the  form 


where  CJv  is  the  thermal  capacity  of  a  unit  volume  of 
the  gas. 

Putting  for  /  its  value  we  have 


TRANSPORT   PROBLEMS.  95 

which,  however,  cannot  be  used  to  compute  either  N 
or  o-  unless  one  of  them  is  previously  known. 

Like  the  viscosity  the  thermal  conductivity  is  pro- 
portional to  the  square  root  of  the  absolute  tempera- 
ture, and  independent  of  the  pressure  or  density., 
Using  the  same  substitution  as  before  the  coefficient 
becomes 

Cv       l~RT~          _i 

2N(T2\'ir*Nm       2(7—  i)er2 

where  7  is  the  ratio  of  the  two  specific  heats. 

The  applicability  of  this  formula  to  actual  gases  is 
subject  to  limitations  similar  to  those  mentioned  in 
connection  with  the  viscosity  of  gases.  It  appears 
that  the  departure  of  the  heat  conductivity  of  rarefied 
gases  from  constancy  is  intimately  associated  with  the 
breaking  down  of  the  condition  that  the  distance 
between  the  solid  surfaces  between  which  the  heat  is 
conducted  is  large  in  comparison  with  the  mean  free 
path  of  the  molecules.  There  must  be  a  slight  dis- 
continuity in  temperature  at  these  surfaces,  since  the 
molecules  just  in  contact  with  each  surface,  instead  of 
being  at  the  temperature  of  the  surfaces,  can  be  re- 
garded as  made  up  of  two  classes ;  one,  of  those 
molecules  which  are  approaching  the  surface,  and 
hence,  coming  from  cooler  portions  of  the  gas,  are  at 
a  lower  temperature  than  the  solid  surface,  and  the 
other  consisting  of  molecules  just  leaving  the  surface, 
and  at  its  temperature ;  the  layer  of  gas  next  to  the 
surface  will  then  consist  of  a  mixture  of  the  mole- 


96  KINETIC   THEORY. 

cules  of  these  two  classes,  and  will  have  an  average 
temperature  slightly  lower  than  that  of  the  solid  sur- 
face it  touches,  or  at  the  other  surface  slightly  higher. 
It  is  possible  to  take  account  of  this  variation  by  intro- 
ducing as  a  correction  to  the  distance  d,  for  each 
surface,  an  amount  equal  to  the  distance  from  the 
geometrical  position  of  the  surface  back  to  the  plane 
where  the  gas  would  have  the  same  temperature  as 
the  surface  on  the  supposition  that  the  temperature 
exhibited  in  that  region  the  same  linear  variation  with 
the  distance  which  it  does  exhibit  between  the  planes. 
This  correction  distance  seems  to  be  proportional  to 
the  mean  free  path,  and  the  ratio  has  been  determined 
for  air  and  hydrogen  by  E.  Gehrcke1  who  finds  that 
for  the  former  the  correction  at  one  surface  is  1.83 
times  the  mean  free  path,  while  for  hydrogen  it  is  5.78 
times  that  distance. 

Diffusion. — This  problem  differs  from  those  just  dis- 
cussed in  two  respects  ;  the  quantity  which  is  carried 
from  one  region  to  the  other  is  composed  of  the  mole- 
cules of  the  gas  itself,  and  as  diffusion  is  ordinarily  of 
one  gas  into  another,  there  are  two  kinds  of  molecules 
present.  We  shall  as  before  take  our  Z-axis  in  the 
direction  in  which  the  diffusion  takes  place,  but  we 
shall  change  our  notation  so  that  the  subscript  I  will 
apply  to  molecules  of  the  first  kind,  and  the  subscript 
2  to  molecules  of  the  second  kind.  Then  at  any  point 
the  number  of  molecules  of  the  first  kind  per  unit 
volume  will  be  nv  the  mass  of  each  molecule  mv  and 

^ Drude* s  Ann.,  2,  p.  112,  1900. 


TRANSPORT    PROBLEMS.  97 

its  velocity  cl  ;  while  the  corresponding  quantities  foi 
the  second  kind  will  be  ;/2,  ;/z2  and  c.r  Let  the  position 
of  the  plane  perpendicular  to  the  direction  of  the  diffu- 
sion be  given  by  its  coordinate,  z,  and  for  the  sake  of 
definiteness  the  positive  direction  be  so  chosen  that  for 
larger  values  of  z  the  density  of  the  first  kind  of  gas 
shall  be  greater,  that  is,  dnjcz  >  o  ;  then  the  oppo- 
site will  usually  be  true  for  the  second  gas,  namely 
dn2jdz  <  o.  The  molecules  of  the  first  kind  passing 
downward  through  the  plane  z  may  be  assumed  as 
before  to  have  come  an  average  distance  /p  but  in 
various  directions,  so  that  they  may  be  spoken  of  as 
coming  from  a  plane  whose  coordinate  is  z  +  ^  cos  $. 
We  have  previously  found  the  average  value  of  this 
expression  to  be  z  -f-  f  /r  If  n^  be  the  number  of  mole- 
cules per  unit  volume  at  the  plane  zt  the  correspond- 
ing number  at  the  plane  z  +  f  /L  will  be 


and  the  number  from  this  plane  passing  downward 
through  unit  area  of  the  plane  z  in  unit  time  will  be, 
supposing  them  all  to  have  the  same  speed  cy 


Similarly  the  number  of  molecules  of  the  first  kind 
passing  upward  through  the  plane  z  in  unit  time  will  be 


98  KINETIC   THEORY. 

and  the  excess  of  the  number  of  those  passing  down- 
ward over  those  passing  upward,  which  is  the  measure 
of  the  rate  of  diffusion  will  be  the  difference  of  these 
two  quantities,  or 

' 


Simply  interchanging  subscripts  and  signs  we  have  the 
-  excess  of  diffusion  of  the  molecules  of  the  second  kind 
upward 


which  is  intrinsically  positive  when 


In  seeking  to  find  the  relation  between  the  theoret- 
ical discussion  and  actual  cases  of  diffusion  we  have  to 
distinguish  between  different  types  of  problems  accord- 
ing as  (a)  the  two  kinds  of  molecules  are  alike  or  dif- 
ferent as  regards  mass,  size  or  any  characteristics  which 
might  affect  their  motion,  (b)  the  total  pressure  exerted 
by  the  two  gases  is  the  same  or  not  in  all  parts  of  the 
region  in  which  the  diffusion  takes  place, 

The  simplest  case  is  that  in  which  the  two  kinds  of 
molecules  differ  in  no  respect  except  that  we  may  sup- 
pose them  capable  of  being  identified  as  to  their  kind, 
as  for  instance  by  a  difference  of  color.  Then  we  will 
have  ml  =  m2,  cv  =£2,  /j  =  /2,  and  for  these  quantities 
may  drop  the  subscripts.  The  pressure  and  tempera- 
ture being  supposed  uniform  throughout  the  space,  the 


TRANSPORT   PROBLEMS.  99 

sum  of  the  number  of  molecules  of  the  two  kinds  per 
unit  volume  will  be  a  constant, 


=  n 


and  the  rate  of  diffusion  of  the  molecules  of  each  kind 
will  be 


the  rates  being  the  same,  but  the  directions  opposite 
for  the  two  kinds,  while  the  gas  as  a  whole  will  remain 
stationary.  The  mass  of  each  kind  of  gas  passing 
through  unit  area  in  one  second  will  be  this  expression 
multiplied  by  m,  the  mass  of  the  molecule,  that  is, 

..  -dn,        „  -dnjn 


m 


dz 


where  the  derivative  dn^m/dz  expresses  the  rate  of 
change  of  the  density  of  the  first  kind  of  gas  along  z. 
The  transformations  effected  by  substituting  for  /  its 
approximate  value  and  introducing  the  temperature  in 
place  of  c  are  entirely  similar  to  those  observed  in  the 
previous  cases,  and  need  no  further  discussion. 

If  we  assume  that  only  one  kind  of  gas  is  present, 
and  that  the  pressure  is  not  uniform,  we  have  n2  =  o, 
and  our  problem  becomes  one  of  dynamics,  the  ques- 
tion of  the  speed  of  a  wind  due  to  a  given  difference 
of  pressure  or  density  ;  if  the  diffusion  is  to  take  place 
through  tubes  or  orifices  of  finite  dimensions,  the  fric- 
tion of  the  gas  against  the  wall  of  the  tubes  will  cause 
the  predominant  phenomena  to  be  those  of  viscosity. 

If  on  the  other  hand  we  attempt  to  solve  the  prob- 


100  KINETIC   THEORY. 

lem  of  the  inter-diffusion  of  two  gases  we  are  con- 
fronted by  two  difficulties  :  the  value  of  /p  the  mean 
free  path  of  a  molecule  of  the  first  kind,  is  certainly  not 
the  same  as  if  only  the  n^  molecules  of  the  first  kind 
were  present,  for  there  is  also  the  probability  of  col- 
lisions with  molecules  of  the  second  kind,  and  it  is 
probably  not  the  same  that  it  would  be  if  all  the  n^  4-  «2 
molecules  were  of  the  same  kind,  and  the  numbers  of 
the  two  kinds  of  molecules  passing  through  any  plane 
in  the  opposite  directions  are  not  necessarily  the  same, 
and  hence  there  must  result  an  inequality  of  pressure 
in  different  regions,  or  else  a  general  drift  of  the  mix- 
ture of  gases  just  sufficient  to  make  up  for  the 'differ- 
ence in  their  diffusions,  the  former  being  observed  in 
the  case  of  the  diffusion  of  gases  through  a  porous 
wall,  as  of  plaster  of  Paris,  while  the  latter  must  occur 
in  free  diffusion. 

The  mean  free  path  for  a  molecule  of  either  kind  in 
a  mixed  gas  made  up  of  two  components  may  be 
found  approximately  as  follows  : 

We  have  found  (p.  60)  that  the  average  number  of 
collisions  per  second  of  a  single  molecule  is 

(23)  P=Trna*r. 

If  we  designate  the  quantities  referring  to  the  impacts 
of  a  molecule  of  the  first  kind  against  molecules  of  the 
second  by  the  subscript  1 2,  this  formula  takes  the  form 

Pu  = 


TRANSPORT   PROBLEMS.  IOI 

where  Pn  is  the  number  of  collisions  with  molecules 
of  the  first  kind,  and  P12  with  those  of  the  second  kind. 
(crj  -f  <72)  /  2  will  be  the  distance  between  the  centers 
of  the  two  molecules  of  different  kinds  at  the  instant 
of  collision.  The  total  number  of  collisions  per  sec- 

ond will  be 

P  —  P   4-  P 
M  —  Mi  -r  *if 

rn  is  identified  with  r  as  previously  found.  r12,  how- 
ever, the  relative  speed  of  two  molecules  of  different 
kinds,  is  still  to  be  found.  If  we  make  the  assump- 
tion that  all  the  molecules  of  the  first  kind  have  the 
same  speed,  cv  and  all  those  of  the  second  the  same 
speed  r2,  the  relative  speed  of  two  molecules  of  differ- 
ent kinds  will  be  given  by  the  equation 


cos 


If  we  take  for  cos  &  the  average  value  o,1  as  if  the 
molecule  of  the  first  kind  were  projected  with  the 
velocity  cl  into  a  region  occupied  by  molecules  of  the 
other  kind  all  moving  with  the  speed  c.2  in  planes  per- 
pendicular to  the  direction  of  the  newcomer,  we  have 


From  the  equilibrium  of  temperature  which  may  be 
assumed  we  have 


1  See  note,  pp.  60-63,  in  which  the  formula  7  =  2  /  TT  V  a2  -f  /32  would 
reduce  to 


102  KINETIC   THEORY. 


which,  if  the  two  kinds  of  molecules  have  the  same 
mass,  reduces  to 

r  =  c\/2, 

one  of  the  forms  previously  obtained.  Inspection 
shows  that  the  values  of  r12  and  r2l,  which  can  be 
obtained  by  interchanging  the  subscripts  are  the  same, 
and  that  when  the  two  kinds  of  molecules  have  very 
different  speeds  and  masses  the  relative  speed,  is  very 
nearly  the  average  speed  of  the  lighter  and  swifter 
molecule  ;  as,  for  instance,  if  I  refer  to  hydrogen  and 
2  to  oxygen, 

wi_.   i 

™    -  16' 


=  1-0308^, 

4.1231^, 

which  are  equal,  as  has  just  been  stated,  since 


The  total  number  of  collisions  per  second  of  a  mole 
cule  of  the  first  kind  then  becomes 


TRANSPORT   PROBLEMS.  103 


and  the  mean  free  path  of  such  molecules 

,  „  f. 

1    P. 


and  their  rate  of  diffusion  will  be 

i/r^1 

*'iC*  dz 


dn. 


i 

1o-12  +  3       i  4- 


an  expression  which  depends  upon  the  sizes,  numbers 
and  relative  masses  of  the  two  kinds  of  molecules  as 
well  as  upon  their  temperature  and  pressure.  Simi- 
larly the  rate  for  molecules  of  the  other  kind  will  be 


These  two  values  will  probably  differ  somewhat,  so 
that  diffusion  at  these  rates  would  result  in  an  excess 
of  pressure  in  one  part  of  the  system  over  that  in 


IO4 


KINETIC   THEORY. 


other  parts,  with  a  consequent  general  drift  of  the 
mixture,  which  is  measured  by  the  difference  of  these 
two  coefficients,  of  which  the  two  kinds  are  present  in 
the  proportions  of  nv  and  ny  so  that  the  total  drift 
will  be 


,—  „  ///„  /  O,     -|-    U 

3T/27r^2o-22+3^i  +-*v*l{- 


and  the  corrected  coefficient  for  the  first  kind  of  mole- 
cules will  be  the  original  coefficient  diminished  by 
njn  times  this  drift,  where  n  =  n^\-nv  which  reduces  to 


^  -f  3 


TRANSPORT   PROBLEMS.  105 

This  formula  may  be  simplified  so  as  to  be  com- 
pared with  experimental  results  in  either  of  the  two 
ways.  Some  have  assumed  that  the  rate  of  diffusion 
depends  only  upon  the  mutual  collisions  of  molecules 
of  different  kinds,  but  not  upon  those  of  molecules  of 
the  same  kinds.  Upon  this  assumption  the  expres- 
sion above  reduces  to 


-f  1/»V2 


which  may  be  reduced,  since  <r2  =  ^  A/—1*  to  the  form 

if  ?^0 


m.— 


5nt 


n  may  be  eliminated  and  the  pressure,  /,  be  intro- 
duced, by  the  equation 

p-^nm^  -.^-nm^cj  , 
which  gives 


106  KINETIC   THEORY. 


According  to  this  formula  the  coefficient  of  diffusion 
in  terms  of  the  change  of  density  of  the  molecules  of 
the  first  kind  is  independent  of  the  relative  propor- 
tions of  the  two  kinds  of  molecules  present,  but  varies 
inversely  as  the  pressure,  and  directly  as  the  |  power 
of  the  temperature.  Expressed  in  terms  of  the  pres- . 
sure  gradient,  which  is  itself  proportional  to  the  tem- 
perature, the  rate  of  diffusion  is 


in  which  the  coefficient  is  proportional  only  to  the 
square  root  of  the  temperature. 

The  assumption  that  the  rate  of  diffusion  depends 
only  upon  the  collisions  between  unlike  molecules 
seems  improbable,  and  justifiable  only  as  a  first  ap- 
proximation. The  final  decision  between  the  special 
formula. just  given  and  the  more  general  one  must  be 
made  on  the  basis  of  experiment,  but  the  discrepan- 
cies in  experiments  so  far  made  are  such  as  to  render 
any  decision  based  upon  them  doubtful.  For  the 
majority  of  such  experiments  a  sufficiently  accurate 
approximation  to  the  general  formula  may  be  made 
by  putting  n^  =  nv  which  gives 


TRANSPORT   PROBLEMS. 


IO/ 


I 

C\ 

39T//- 

V~2*?  +  - 
-4-  

/            ^i/^i+^V 

v  f^A   2  J 

A  sharp  discrimination  might  be  made  by  testing  the 
diffusion  when  only  a  very  slight  amount  of  one  of  the 
gases  present,  when  the  diffusion  will  approach  the 
limiting  value  for  n2  =  o,  which  is 


For  a  more  exhaustive  discussion  the  reader  is 
referred  to  the  work  of  O.  E.  Meyer.1  The  notation 
there  employed  is  quite  different,  and  the  constant 
obtained  is  slightly  different,  8  appearing  in  the  place 
of  37T  in  the  last  formula  given. 


"The  Kinetic  Theory  of  Gases,"  pp.  247-276. 


CHAPTER   V. 
CHANGE   OF   STATE. 

ON  account  of  the  greater  simplicity  of  the  phe- 
nomena of  the  gaseous  state,  the  formulae  connecting 
these  phenomena  and  the  kinetic  theory  for  this  state 
were  developed  much  earlier  and  more  rapidly  and 
completely  than  the  corresponding  formulae  and 
theory  for  the  liquid  and  solid  states. 

Clausius  early  pointed  out  some  of  the  general  con- 
siderations which  must  lie  at  the  basis  of  a  kinetic 
theory  of  liquids,  and  in  particular  gave  us  a  sort  of 
picture  of  the  processes  which  must  take  place  in  the 
vaporization  of  a  liquid  or  in  the  condensation  of  a 
vapor ;  the  way  in  which  these  processes  take  place 
we  have  attempted  to  describe  very  briefly  at  the  end 
of  the  introduction.  Since  we  have  a  fairly  definite 
kinetic  theory  of  gases,  this  transition  between  the 
liquid  and  gaseous  state  has  seemed  to  furnish  a  par- 
ticularly available  method  of  finding  out  what  is  the 
behavior  of  the  molecules  of  a  liquid.  In  order  that 
we  may  do  this  with  the  greater  confidence,  we  shall 
here  review  some  of  the  familiar  facts  regarding 
change  of  state  and  deduce  briefly  from  thermodynam- 
ical  considerations  some  equations  which  we  may  find 
interesting  or  useful. 

In  the  case  of  an  ordinary  liquid,  such  as  water,  if 
108 


CHANGE  OF   STATE.  109 

heat  be  applied  to  it,  there  results  a  slight  expansion, 
and  increase  of  temperature,  and  perhaps  some  slight 
vaporization  from  the  surface,  which  however  may  be 
prevented  by  having  the  liquid  confined  so  that  there 
is  no  free  space  above  it ;  if  the  pressure  is  kept  con- 
stant and  heat  still  added  a  temperature  is  reached  at 
which  vapor  tends  to  form  not  simply  at  the  surface 
of  the  liquid  but  also  in  bubbles  within  the  body  of  the 
liquid  ;  if  still  more  heat  is  added,  and  the  pressure 
still  kept  constant,  vaporization  continues,  with  great 
increase  of  volume  but  no  accompanying  rise  of  tem- 
perature until  the  whole  liquid  is  vaporized  ;  after  that, 
further  addition  of  heat  results  in  rise  of  temperature 
and  increase  of  volume  as  in  any  gas.  A  more  com- 
plete and  systematic  study  can  be  made  by  plotting 
the  volume  of  the  substance  as  the  abscissa,  and  its 
pressure  as  ordinate,  and  drawing  the  isothermal 
curves  for  different  temperatures.  A  good  example  is 
the  classical  work  of  Andrews  on  carbon  dioxide,  the 
diagram  for  which  is  found  in  most  text-books  on 
heat.  He  found  that  the  isothermals  for  this  substance 
were  of  two  kinds,  those  for  higher  temperatures  being 
curved  throughout  their  whole  length  and  of  varying 
steepness,  but  never  horizontal,  while  below  a  certain 
temperature  they  seemed  to  consist  of  three  parts 
characteristically  different,  a  veiy  steep  part  which 
related  to  the  substance  in  its  liquid  state,  another 
curved  part  which  related  to  the  vapor  state,  and 
between  them  a  straight  horizontal  portion,  which  re- 
ferred to  the  substance  when  partly  in  the  liquid  and 


no 


KINETIC   THEORY. 


partly  in  the  vapor  state.  We  may  draw  through  the 
ends  of  these  straight  portions  where  they  join  the 
part  of  the  curve  corresponding  to  the  liquid  state  a 
curve  which  we  shall  call  the  "  water  line,"  and  simi- 
larly through  the  other  ends  of  these  straight  portions 
a  curve  which  we  shall  call  the  "steam  line." 


100 


90 


V 


60 


60 


C.C.   PER  GRAM 
Fig.    8. 

Andrews'  experiments  and  many  others  show  that  the 
water  line  and  steam  line  join  at  a  point  which  is  com- 
monly known  as  the  CRITICAL  POINT.  The  isother- 
mals  above  this  point  have  no  horizontal  part,  those 
below  have  the  portion  between  the  water  and  steam 
lines  horizontal,  while  the  isothermal  through  the 


CHANGE  OF  STATE.  Ill 

critical  point  has  at  that  point  a  point  of  inflection 
where  it  is  horizontal  also,  but  it  is  not  so  at  any  other 
point  of  its  whole  length.  All  these  properties  may 
be  shown  by  means  of  a  surface,  which  has  for  its 
three  coordinates  the  pressure,  volume  and  tempera- 
ture of  the  substance.  The  accompanying  figure  is 


F!g.  9. 

reproduced  from  a  photograph  of  a  model  of  such  a 
surface,  which  belongs  to  the  University  of  California  ; 
in  it  the  temperature  is  measured  vertically  upward, 
so  that  a  horizontal  section  gives  an  isothermal  curve. 
As  the  figure  is  placed  pressure  is  measured  positively 
to  the  left,  and  volume  forward,  away  from  the  observer. 
The  portion  of  the  surface  which  appears  nearly  plane 
represents  the  area  of  the  diagram  between  the  water 


112  KINETIC   THEORY. 

and  steam  lines,  corresponding  to  the  coexistence  of 
the  liquid  and  vapor  phases  of  the  substance.  A  more 
accurate  representation  of  the  facts  would  have  made 
this  part  of  the  surface  strongly  convex  upward,  as  a 
vertical  section  parallel  to  the  axis  of  pressures  gives 
the  curve  of  the  tension  of  the  saturated  vapor,  in 
which  the  tension  is  found  by  experiment  to  increase 
more  rapidly  than  the  temperature. 

Thermodynamics. —  If  we  call  the  volume  of  the  sub- 
stance in  the  liquid  state  vl  and  in  the  vapor  state  v2 
and  if  the  volume  occupied  by  it  when  partially  vapor- 
ized is  Vy  then 


is  the  fraction  or  proportion  of  the  substance  which 
has  been  vaporized,  and 


i  —  m  = 


is  the  fraction  remaining  in  the  liquid  state,  m  as 
thus  defined  can  have  all  values  between  o  and  I,  but 
has  no  meaning  except  when  the  two  phases  are  coex- 
istent, or  in  the  area  between  the  water  and  steam 
lines.  We  may  solve  for  v,  getting 

v  =  ( i  —  in)i\  -f  mVy 

For  convenience  we  may  call  the  specific  heat  of  the 
liquid  at  constant  m,  that  is  the  amount  of  heat  re- 
quired to  raise  the  temperature  of  unit  amount  one 
degree,  keeping  it  at  such  a  pressure  that  it  is  just 


CHANGE   OF   STATE.  113 

ready  to  vaporize,  Cm,  and  similarly  the  specific  heat 
of  the  vapor  when  just  saturated  Hm.     The  latent 
heat  of  vaporization  we  shall  call  L,  that 
is,  the  amount  of  heat  required  to  just  va- 


porize the  unit  amount  of  liquid,  keeping 

A  11  Fte«  10. 

its  temperature  constant.      All  amounts 

of  heat  and  energy  we  shall  suppose  measured  in  the 

same  kind  of  units,  for  convenience. 

Consider  now  a  small  Carnot  Cycle,  of  length  dv 
and  height  dp.  Its  area  is  then  the  product  dpdvy  or 
substituting  for  dv  and  dp 

dv  =  (v2  —  v^)  dm 


the  last  equation  being  true  since  the  pressure  of  the 
vapor  depends  upon  the  temperature  only,  we  have 

for  the  area 

dp 
dpdv  =  (v2  -—  z/j)  dm  -^dT. 

Now  this  expression,  the  area  of  the  cycle  on  the 
diagram  of  pressure  and  volume,  is  equal  to  the  work 
done  by  the  substance  if  caused  to  pass  through  this 
cycle.  The  heat  absorbed  by  the  substance  in  the 
expansion  dv  is  entirely  used  in  vaporizing  the  amount 
dm  of  the  liquid  and  is  hence  equal  to 

Ldm. 

Now  it  is  easily  shown,  and  is  a  fundamental  truth  of 
thermodynamics  that  the  efficiency  of  a  Carnot  Cycle, 
8 


114  KINETIC   THEORY. 

that  is  the  ratio  of  the  work  done  to  the  heat  taken  in 
at  the  upper  temperature,  is  equal  to  the  ratio  of  the 
range  of  temperature  to  the  upper  temperature.  Ap- 
plying this  theorem  to  this  case  we  have 


Ldm  '   T 

whence 

(29)  Z=7K-^W~- 


We  can  easily  write  the  equations  corresponding  to 
those  we  have  given  for  gases.  For  instance,  the  first 
law  of  thermodynamics  becomes 


dQ  =  [(i  -  m)Cm  +  mH^dT+  Ldm 
(30) 


Special  cases  of  interest  are  : 

The  isothermal  expansion,  dT=  o,  /  =  const, 

dQ  =  Ldm. 
Expansion  with  m  kept  constant 


Special  cases  : 

m  =  o,  liquid  state  -dQ  —  CmdT, 
m=  i,  vapor  state  .  dQ  =  HmdT. 

Adiabatic   Expansion  and  Entropy.  —  We    may  write 
the  equation  expressing  the  first  law  in  the  form 

=  TdS 


CHANGE   OF   STATE.  115 

where 

Jf-m(ffm-CJ+Cm. 
Then 


for  inserting  these  values  the  equation  becomes  an 
identity. 

Differentiating  these  partially  by  m  and  T  respec- 
tively and  taking  their  difference 

dL      dM     dS 


dT     dm  ~~  dm  7"  T dmdT      7  dmdT 

dS      L        .       dT 

—  *~  =  ~^r     (since  ~—  =  o). 
*  dm      T     v          30* 

But  substituting  the  value  of  M 


-H       C 
dm       a»      U 


or 

H  -  C  =  a-- -. 

Substituting  this  value  the  first  law  becomes 

(3 1)  dQ  =  m  (  ^  —  -=, \  dT+  CmdT+  Ldm  =  TdS. 

Then 

i  dL       L  \  L  ,  dT 


KINETIC   THEORY. 

-  dT 


u6 


and  if  Cm  can  be  regarded  as  constant,  which  it  probably 
is  very  nearly, 

(32)  5  =  ^  +  Cm  log  T  +  const. 

and  the  equation  of  an  adiabatic  is 

mL 

-~r  -f  Cm  log  /  =  const. 

To  write  these  last  equations  in  terms  of/  and  v  would 
require  definite  knowledge  of  the  relation  between  T 
and  /,  a  relation  which  is  not  easy  to  deduce  on  theo- 


Fig.  11. 


retical  grounds,  and  does  not  yet  appear  to  be  easily 

and  satisfactorily  generalized  from  experimental  data. 

It  was  early  suggested  by  Professor  James  Thomson, 


CHANGE  OF  STATE.  II 7 

brother  of  Lord  Kelvin,  that  besides  the  actual  forms 
of  the  isothermals,  there  might  be  an  ideal  form,  which 
should  not  have  the  straight  line  nor  the  sharp  bends 
of  the  form  we  have  been  discussing.  The  general 
form  he  suggested  is  shown  in  the  diagram.  The  form 
of  the  surface  or  model  would  be  different  from  that 
shown  in  Fig.  9,  having  at  the  left,  over  part  of  the 
area  representing  the  change  of  state,  a  ridge,  and  to 
the  right  a  deep  hollow.  A  vertical  section  parallel 
to  the  axis  of  volume  would  have  somewhat  the  same 
general  shape  as  an  isothermal,  only  reversed,  show- 
ing with  increasing  volume  the  temperature  first  rising, 
then  falling  and  then  again  rising.  Points  on  the  left 
side  of  the  ridge,  or  the  right  side  of  the  hollow  might 
refer  to  states  actually  attainable  ;  for  instance,  a  liquid 
can  be  heated  above  its  ordinary  temperature  of  boiling 
without  boiling  at  all,  until  finally  it  begins  to  do  so 
almost  explosively  ;  this  explosive  action  would  seem  to 
be  the  accompaniment  of  a  passage  from  one  state  to  the 
other,  which  might  be  represented  by  a  passage  from 
one  portion  of  the  surface  to  the  other.  The  two  iso- 
thermals as  we  have  seen  evidently  do  not  belong  to  the 
same  surface,  the  actual  isothermal  corresponding  to 
the  actual  changing  from  the  liquid  to  the  vapor  state 
by  vaporization,  a  discontinuous  process  during  which 
the  substance  is  present  in  two  phases,  while  the  ideal 
isothermal  corresponds  to  a  continuous  passage  from 
one  state  to  the  other,  a  purely  hypothetical  process, 
during  which  the  substance  has  only  one  phase.  The 
points  on  the  isothermal  at  which  it  slopes  down  from 


Il8  KINETIC   THEORY. 

left  to  right  refer  to  conditions  which  may  possibly  be 
attainable  when  for  instance  a  liquid  is  superheated,  or 
a  vapor  undercooled ;    but  the  middle  region,  where 
the  slope  is*  upward,  represents  an  unstable  and  explo- 
sive state,  in  which  an  increase  of  volume  is  accom- 
panied by  an  increase  of  pressure.     This  may  perhaps 
be  regarded  as  the  reason  why  the  actual  passage  from 
one  state  to  the  other  is  discontinuous,  each  little  part 
of  the  liquid,  as  it  receives  sufficient  heat,  passing  sud- 
denly and  explosively  to  the  vapor  state,  thus  creating 
the  two  phases.      One  consideration  of  considerable 
importance  can  be  stated  ;  the  fact  that  the  amount  of 
work  a  body  can  do  during  a  closed  cycle  as  the  result 
of  an  amount  of  heat  given  to  it  is  proportional  to  the 
range  of  temperature  of  the  cycle  leads  to  the  corollary 
that  no  work  can  be  done  by  an    isothermal   cycle. 
This  might  indeed  be  stated  as  one  of  the  forms  of  the 
Second  Law  of  Thermodynamics.     If  now  a  body  of 
liquid  be  isothermally  vaporized  completely,  and  then 
at  the  same  temperature  caused  to  return  to  the  orig- 
inal liquid  state  by  the  continuous  process  according 
to  the  ideal  isothermal,  then  in  the  whole  cycle  no 
work  is  done,  and  hence  algebraically  the  sum  of  all 
the  areas  on  the  plane  of  pressure  and  volume,  which 
represents  the  work  of  the  cycle,  must  be  zero.     The 
process  is  a  purely  hypothetical  one,  but  the  logic  is 
conclusive,  and  we  learn  from  it  that  the  two  areas 
enclosed  between  the  two  isothermals  are  equivalent. 
This  relation  we  shall  have  occasion  to  make  use  of 
later.     The  fact  that  there  is  a  critical  point  above 


CHANGE   OF   STATE.  IIQ 

which  the  distinction  between  liquid  and  vapor  ceases, 
and  that  a  vapor  may  be  made  to  pass  from  the  gase- 
ous to  the  liquid  state  continuously  by  compressing 
it  at  a  temperature  above  the  critical  temperature,  and 
then  cooling  it,  carefully  keeping  its  volume  less  than 
that  at  which  it  can  begin  to  vaporize,  seem  to  add 
significance  to  this  idea  of  an  ideal  isothermal  of  con- 
tinuous change  of  state. 


CHAPTER  VI. 
EQUATION   OF  VAN   DER  WAALS. 

So  long  as  the  actual  bulk  of  all  the  molecules  of 
a  substance  is  negligible  in  comparison  with  the  volume 
occupied  by  the  substance,  and  so  long  as  the  mutual 
forces  between  the  molecules  only  act  for  distances  so 
short  in  comparison  with  the  mean  free  paths  of  the 
molecules  that  the  portions  of  their  paths  which  are  not 
straight  are  negligible  in  comparison  with  the  straight 
portions,  the  substance  will  follow  the  laws  of  Boyle 
and  Charles,  and  will  be  properly  called  an  ideal  gas, 
whether  the  molecules  be  smooth,  hard  spheres,  or  of 
less  simple  shapes,  or  even  aggregations  more  or  less 
complex,  and  whether  the  rebound  be  due  to  simple 
elastic  forces  on  contact,  or  to  forces  acting  as  some 
power  of  the  distance  between  the  centers  of  the  mole- 
cules. But  no  actual  substances  follow  exactly  the 
laws  of  Boyle  and  Charles,  and  hence  the  necessity 
of  a  theory  more  general.  The  equation  of  van  der 
Waals,  which  we  are  about  to  discuss,  is  more  general 
in  two  respects  :  first,  it  relieves  us  of  the  restriction 
that  the  volume  of  the  molecules  of  the  substance  is 
negligible  ;  second,  it  makes  the  assumption  that  there 
exist  between  the  molecules  forces  of  mutual  attrac- 
tion, which  we  may  call  cohesive  forces.  The  second 
assumption  is  one  which  can  evidently  apply  to  liquids, 

120 


EQUATION   OF   VAN    DER   WAALS.  121 

and  it  would  appear  that  it  must  apply  to  gases  also, 
since  all  known  gases  can  be  liquefied,  and  the  same 
substance  can  exist  at  the  same  temperature  and  pres- 
sure side  by  side  in  the  two  phases  of  liquid  and  vapor. 
We  can  give  this  assumption  more  definite  form  by 
saying  that  besides  the  repulsive  forces  which  act  very 
strongly,  but  only  at  very  small  distances,  each  mole- 
cule of  the  substance,  whether  that  substance  be  liquid 
or  gas,  exerts  upon  each  other  molecule  of  the  same 
substance  a  force  which  falls  off  with  the  distance,  but 
slowly  enough  so  that  at  a  considerable  distance  the 
force  may  be  regarded  as  constant ;  then  any  mole- 
cule of  the  substance  is  subject  to  attractive  forces 
exerted  by  molecules  in  all  directions  from  it,  which 
if  it  be  well  within  the  body  of  the  substance  are  sen- 
sibly in  equilibrium  and  cannot  seriously  affect  its 
motion  ;  but  a  molecule  near  the  surface  of  this  sub- 
stance is  subject  to  these  forces  on  the  side  toward  the 
body  of  the  substance,  but  only  to  a  very  much  less 
degree  on  the  opposite  side,  the  less  as  it  approaches 
nearer  the  surface.  If  now  the  substance  be  restrained 
by  the  walls  of  the  containing  vessel,  it  is  subject  to 
two  forces,  one  /,  the  pressure  exerted  by  the  walls 
themselves,  which  could  be  measured  by  a  pressure 
guage,  such  as  a  column  of  mercury,  the  other,  Pt 
the  cohesive  force  due  to  the  mutual  attraction  of  the 
molecules.  The  whole  force  acting  upon  the  surface 
of  the  substance,  which  results  in  the  returning  to  the 
interior  of  the  substance  of  any  molecules  starting  out 
toward  the  surface  is  the  sum 


122  KINETIC   THEORY. 


The  form  of  the  expression  for  the  molecular  pressure 
was  deduced  by  van  der  Waals  in  the  following  man- 
ner :  our  lack  of  knowledge  of  the  law  of  the  forces 
between  molecules  would  be  a  very  serious  matter  if 
we  were  dealing  with  the  motions  of  one  molecule 
only,  but  as  we  are  dealing  with  great  numbers  of 
them,  we  may  be  satisfied  with  average  results  ;  then 
on  the  average  the  force  acting  on  any  molecule  near 
the  surface  tending  to  draw  it  back  within  the  sub- 
stance is  proportional  to  the  density  of  the  substance, 
that  is,  to  the  number  of  attracting  molecules,  and  the 
total  force  exerted  upon  a  definite  portion  of  the  sur- 
face is  again  proportional  to  the  number  of  molecules 
in  it,  that  is,  to  the  density  of  the  substance,  hence 
the  total  molecular  pressure  is  proportional  to  the 
square  of  the  density  of  the  substance,  or  to  the 
inverse  square  of  the  specific  volume,  that  is 


The  considerations  which  we  have  adduced  in  Chapter 
III.  with  regard  to  the  effect  upon  the  pressure  of  the 
volume  of  the  molecules  also  apply  here,  and  hence 
our  equation  takes  the  form 

(33) 

In  our  deduction  so  far  we  have  carefully  used  the 
word  substance,  since  everything  we  have  said  may  be 


EQUATION   OF   VAN-  DER   WAALS.  123 

equally  well  stated  of  either  liquids  or  gases.  In  fact, 
van  der  Waals  was  led  to  this  form  of  discussion  by 
his  study  of  the  theories  of  capillarity  and  surface  ten- 
sion ;  so  that  it  would  seem  to  be  a  fair  suggestion 
that  this  equation  might  apply  both  to  the  gaseous 
and  to  the  liquid  state.  While  this  formula  does  not 
represent  with  perfect  accuracy  the  behavior  of  actual 
gases,  it  may  be  regarded  as  a  second  approximation, 
much  nearer  the  actual  truth  than  the  first  approxi- 
mation 
(3)  pv  =  RT. 

Other  forms  of  this  equation  have  been  proposed,  as 
that  of  Clausius, 


or  of  Dieterici 


For  the  present,  however,  we  shall  confine  our  atten- 
tion to  van  der  Waals'  form  of  the  equation,  and  to  that 
simply  as  describing  the  phenomena  of  a  substance  in 
the  gaseous  state.  The  more  complete  discussion  of 
its  properties  we  shall  postpone  to  a  later  part  of  the 
chapter. 

A    gas    satisfying    this    equation    does   not   follow 
Boyle's  Law.     For  we  may  write 

i  Wied.  Ann.,  69,  p.  685,  1899. 


124  KINETIC   THEORY. 

RT        a 


RT        a 


MS  iX       —  j  7 

b       v 
I  —  - 
V 

which  is  evidently  not  a  constant.  Whether  the  value 
of  the  product  will  be  greater  or  less  than  RT,  and 
whether  it  will  increase  or  decrease  with  increasing 
volume  depends  upon  the  values  of  a  and  b,  and  can 
only  be  determined  by  experiment. 

Experiments  have  been  made  on  this  point  by 
Amagat,  the  results  of  which  for  air  are  shown  by  the 
accompanying  diagrams,  in  which  the  abscissa  repre- 
sents not  the  volume  but  the  pressure,  and  the  ordi- 
nate  the  product,  pv.  The  broken  line  represents  the 
values  computed  by  van  der  Waals l  from  his  equation. 
The  agreement  is  quite  close,  both  curves  showing  a 
marked  minimum  in  the  va.lue  of  pv.  For  hydrogen 
a  may  be  taken  as  zero,  giving 

RT 

I  —  - 

v 

which  has  no  minimum  point. 

We  may  also  find  how  nearly  such  a  gas  follows 
the  laws  of  Charles  and  Gay  Lussac.  For  an  ideal 
gas  the  coefficients  of  change  of  pressure  and  of  volume 
are  the  same  and  given  by  the  formula 


Continuitat,"  I.,  p.  106. 


EQUATION   OF   VAN    DER   WAALS.  12$ 


where  TQ  is  the  standard  temperature  of  melting  ice, 
about  273°.    For  a  gas  following  van  der  Waals'  equa- 


100   200   300   403   5CO   600   TOO   800    &OO  10OO 

Fig.  12a. 

tion,  at  the  two  temperatures  7]  and  T2  we  may  write 
for  the  pressures  at  the  same  volume 


i.ooo 


980 


.950 


50  100 

Fig.  12b. 


150 


a 


126  KINETIC   THEORY. 


A  —A  =  ^_^&(T2  -  ri) 
A~A  =  __^_ 

That  is,  the  rate  of  change  of  the  pressure  when  the 
volume  is  kept  constant  and  the  temperature  changes 
is  constant,  depending  however  on  the  volume.  The 
coefficient  is  this  rate  divided  by/0,  the  pressure  at  this 
volume  and  at  the  temperature  T0,  which  is 

.  KT0       a 


f»     v—6 

A-A_    R 

A -A 


TQ 


This  pressure  coefficient  is  a  constant,  but  slightly  larger 
than  that  for  an  ideal  gas.  It  becomes  identical  with 
the  latter  for  gas  for  which  a  =  o,  as  hydrogen,  and 
approaches  it  very  closely  for  an  attenuated  gas,  for 
which  v  is  very  large. 

The  volume  coefficient  cannot  be  so  simply  deter- 
mined, as  its  exact  deduction  involves  the  solution  of  a 
cubic  equation,  but  a  useful  approximation  applicable 
to  the  gaseous  state  can  be  made  as  follows  : 

Inasmuch  as  ajv2"  is  a  small  correction,  we  may  sub- 


EQUATION    OF   VAN    DER   WAALS.  I2/ 

stitute  in  it  for  v  its  approximate  value  RT/p.     We 

then  have 

a  _     ap* 

-i 


RT          RT         i  RT(          ap 


ap  p  ap 

~r  ~r>2T2  *  "i    'JyT'TZ 


_RT 
~~^p~' 

Differentiating  by  7",  considering  /  constant 

dv__R         a     _R( 
dT** p*  RT**~ p\  ^R 

but  at  the  temperature  T0 


hence 

I    Bit  I 

v      if  fiT       T 
or  approximately 


128  KINETIC  THEORY. 

I    /  a  a         b 


At  very  high  temperatures,  when  T  is  very  large,  this 
approaches  the  value 

ap 


which  is  a  constant. 

In    the    case  of  hydrogen,   for  which  a  =  o,  our 
formula  becomes 


We  see  that  for  a  gas  for  which  a  =  o,  whose  equa- 
tion is 


This  is  actually  the  case  with   hydrogen,  while  all 
other  gases  have 

a»  >  <V 

Thermodynamics.  —  In  the  case  of  a  gas  which  fol- 
lows van  der  Waal's  equation  we  can  no  longer 
assume  that  the  energy  of  the  gas  is  independent  of 
the  volume,  but  if  it  expands  at  constant  pressure,  the 
amount  of  energy  expended  in  the  expansion  is  meas- 
ured not  by 


but  by 

/(>+$)* 


EQUATION   OF   VAN    DER   WAALS.  129 

But  of  this  second  integral  only  the  first  part  repre- 
sents work  done  by  the  gas  upon  outside  bodies,  the 
second  part  represents  the  increase  of  the  potential 
energy  of  the  gas.  The  first  law  of  thermodynamics 
becomes  then  for  such  a  gas 


in  which  evidently 

r  -  d—      -- 
C*=  dT'      v*~''   dv 

This  equation  may  also  be  written 

7?  T 
(34)  JQ^CjlT+^fidv, 

a  form  which  does  not  contain  /  at  all.  The  other 
forms  of  the  law,  omitting  either  v  or  T,  may  be  ob- 
tained by  elimination  with  van  der  Waals'  equation, 
and  by  the  use  of  the  approximations  we  employed 
in  finding  the  coefficient  of  expansion. 


(33) 


-  J  +  dv  =  RdT, 

omitting  2ab\v"  since  both  a  and  b  are  very  small, 

RdT-  (v  -b}dp  . 
dv  •=•  - 

a 


130  KINETIC   THEORY. 

a 


+         dv  =  -     \RdT-  (v 


Putting  as  before 

a         ap* 


_ 

' 


a  a 

v*  f+v*     RT 


RT  RT (  m    t     ap 

from  which 


RT 
—j- 

and  henee 


or  putting 
(35) 


EQUATION    OF   VAN    DER   WAALS  131 

(36)         dQ  =  CdT  -^(i  + 
Similarly  we  may  eliminate  Tt 

^-J  v-b 


which  may  be  reduced  to  the  form 

(37)       dQ  =f(*-  *X/  +  §  (/  -  5)*' 


The  entropy  of  such  a  gas  is  easily  found,  dividing 
both  members  of  the  equation 


by  T  we  have 

dQ  dT  dv 

—  =dS=C,-T  +  R-^rb, 

and  integrating 

(38)      5  =  Cv  log  T  +  R  log  (v  —  ff)  +  const, 

which  is  the  same  as  the  expression  for  the  entropy 
of  an  ideal  gas,  with  the  co-volume,  v  —  b,  replacing 
the  volume. 

By  the  elimination  of  T  the  equation  of  an  isentropic 
or  adiabatic  line  is  found  to  be 


132  KINETIC   THEORY. 

(\  7? 

/  +  ^J(*>-£)1+^ 


const, 

which  when  a  and  b  are  both  zero  reduces  to  the 
familiar  form  for  an  ideal  gas 

cp 
(21)  pv^=  const. 


Following  is  a  tabulation  of  the  principal  equations 
relating  to  an  ideal  gas,  and  to  one  following  van  der 
Waals'  equation. 

IDEAL  GAS. 

(3)  pv  =  RT, 

i 

*o 
I 

•^o 
dU 


dU 


(2  X)  5  =  C9  log  T  +  ^  log  v  +  Const, 

o> 
(20)  pvd  =  Const., 

RT 

dQ=  C,dT+-^-dv, 

~dp, 


EQUATION   OF   VAN   DER  WAALS.  133 

Cp  Cv 

VAN  DER  WAALS'. 
(33) 


ap  ap         b 


a 


(35)  ..Cp=C9  +  R+2-, 

(38)     5  =  C,  log  T  +  R  log  (v  —  b)  +  Const, 


(39)  /  +         ("-*)^=  Const., 

7?  7"1 
(34) 

(36)  dQ 


(37)        «- 

Ratio  of  Specific  Heats.  —  From  the  discussion  on 
p.  1  30  it  would  appear  that  for  a  gas  which  follows  van 


134  KINETIC   THEORY. 

der  Waals'  equation  the  specific  heat  at  constant  pres- 
sure, Cp,  includes  not  simply  the  heat  necessary  to 
increase  the  kinetic  energy  of  the  molecules,  which 
we  call  Cv,  and  that  necessary  to  do  the  external  work 
accompanying  the  expansion  but  also  the  work  neces- 
sary to  increase  the  potential  energy  of  the  gas,  due 
to  the  intermolecular  attractions.  This  appears  from 
the  formula  /  2ap  \ 

(35)  C*=C*  +  R\l+T&T*] 

\  / 


Determinations  l  of  the  value  of  the  ratio  of  the  two 
specific  heats  are  usually  made  by  methods  involving 
the  adiabatic  expansion  or  compression  of  a  gas,  as  the 
method  by  the  velocity  of  sound,  or  the  change  of 
pressure  on  sudden  rarefaction  or  compression.  Now 
the  equation  of  an  adiabatic,  as  we  have  seen,  was 


(39)  /  +    i(»-*)*  =  Const; 

hence  the  ratio  determined  is  very  nearly 
C  R 


although  the  effect  of  the  term  a/v*  may  be  to  increase 
it  somewhat.  But  the  true  value  of  the  ratio  is  shown 
by  the  equation  above  to  be 

£.  R 


*  Phys.  Rev.,  XII.,  pp.  353-358,  1901. 


EQUATION   OF   VAN   DER  WAALS.  135 

In  our  deduction  of  this  ratio  in  Chapter  IH.,  we  took 
into  account  the  energy  of  the  rotational  motions  of 
the  molecules,  but  not  the  work  of  overcoming  the 
mutual  attractions  of  the  molecules.  Introducing  this 
element,  the  formula 


becomes 

£-i+»J 

Cx_  -" 


This  may  help  to  account  for  the  values  of  Cp/Cv, 
which  were  slightly  higher  than  those  indicated  by 
the  incomplete  theory. 

Form  of  Isothermals.  —  The  equations  of  the  isother- 
mals  of  a  substance  following  van  der  Waals'  equation 

(33) 

are  obtained  by  making  T  constant.  It  is  convenient 
for  the  purpose  of  studying  them  to  arrange  the  terms 
according  to  the  powers  of  v.  Clearing  of  fractions 
and  multiplying  up  we  have 

/z/3  —  pbv21  +  av  —  ab  —  RTv*  =  o, 

and  arranging  according  to  powers  of  v  and  dividing 
by  the  coefficient  of  iP 

a         ab 


This  is  an  equation  of  the  third  degree  in  v,  and  as 
such  has,  according  to  the  theory  of  equations,  three 


136 


KINETIC   THEORY. 


roots,  which  may,  according  to  the  values  of/  and  Tt 
be  all  real,  or  one  real  and  two  conjugate  complex  or 
imaginary  quantities.  In  other  words,  a  horizontal 


\ 


\ 


Fig.  13. 


line,  that  is,  a  line  of  constant  pressure,  may  cut  the 
isothermal  in  one  point  or  in  three,  Following  out 
this  suggestion  we  find  that  the  isothermals  of  such  a 


EQUATION   OF   VAN   DER   WAALS.  137 

substance  have  forms  varying  according  to  the  tem- 
perature from  very  nearly  those  of  an  ideal  gas  to 
forms  resembling  those  suggested  by  Professor  James 
Thomson  for  a  liquid  and  its  vapor,  shown  in  Fig.  1  1  . 
From  the  considerations  there  adduced,  we  are  able 
to  draw  some  conclusions  as  to  the  relative  position 
on  these  isothermals  of  the  straight  lines  representing 
the  actual  transitions  from  the  liquid  to  the  vapor 
state.  If  we  call  the  three  roots  of  our  equations, 
vv  vy  vy  tnen  these  are  the  abscissas  of  the  three 
intersections  of  the  isothermal  by  a  line  of  constant 
pressure.  This  line  of  constant  pressure  will  be  the 
isothermal  corresponding  to  the  actual  transformation 
at  that  temperature,  if  the  area  between  it  and  the 
curved  isothermal  is  algebraically  zero,  or  the  area 
under  each,  between  the  ordinates  vl  and  vy  is  the 
same,  that  is,  calling  the  value  of  the  constant  pres- 
sure P 


S*»3 

Pdv  =  I    pdv. 

Jvi 


Integrating  the  left  side,  and  substituting  for  p  its  value 
R1        a 


RT 


or  dividing  by  (z/3  —  ^), 


138  KINETIC   THEORY. 

RT 


Further,  the  points  P,  vv  and  P,  vy  are  on  the  curve 
and  hence  satisfy  the  equations 

RT 


Z.  "~~         2  ' 

—  b      v* 


•*  7  2 

vs  —  b      vf 

three  equations  which  are  sufficient  to  determine  the 
corresponding  values  of  P,  vlt  and  vy  if  we  were  able 
to  solve  them,  as  functions  of  T. 

Critical  Point.  —  By  the  theory  of  equations,  if  v19 
vv  vy  are  the  three  roots  of  the  equation 

I  RT         \  a         ab 

v  -  \  ~r  +  b  )  v  +  *v  -  -I  =  °- 

V  p       )       P      p 

Then 

RT 


ab 

j- 

Now  at  the  critical  point  the  horizontal  actual  iso 
thermal  vanishes,  shortening  until  its  two  ends  coin 
cide,  and  hence  at  that  point 


EQUATION   OF   VAN    DER   WAALS.  139 

If  we  then  call  the  critical  volume,  pressure  and  tem- 
perature ve,  pc,  Tc, 


RT 


Solving  we  have 


a 


That  is,  the  properties  of  the  critical  state  are  com- 
pletely determined  by  the  constants  a  and  b  of  the 
gas.  The  same  results  would  have  been  obtained  by 
defining  the  critical  point  as  one  at  which  the  iso- 
thermal is  horizontal  and  vhas  a  point  of  inflection,  that 
is,  the  first  and  second  derivatives  vanish.  Its  coordi- 
nates then  satisfy  the  three  equations 

RT        a 

P   =    —       -  7    --  9> 

v  —  b      v2 


RT         2a 


140  KINETIC   THEORY. 

The  solution  of  these  equations  will  give  the  same 
values  of  the  critical  volume,  pressure  and  tempera- 
ture as  before. 

Corresponding  States.  —  Let  us  introduce  as  a  new 
set  of  variables  the  ratios  between  the  actual  and 
the  critical  volumes,  pressures  and  temperatures  which 
we  may  call  the  reduced  volume,  pressure  and  tem- 
perature and  designate  them  by 

(40)  ^1        *-£        »V£ 

Then 


Substituting  these  values  in  the  original  equation  we 
have 


which  reduces  to  the  form 
(41) 


This  is  a  purely  numerical  equation  between  the  re- 
duced volume,  pressure  and  temperature,  entirely  inde- 
pendent of  a,  b  or  R.  (Fig.  1  3  is  plotted  from  this 
equation.)  Two  substances  having  the  same  values 


EQUATION   OF   VAN   DER  WAALS.  141 

of  <f>,  TT  and  #  are  said  to  be  in  CORRESPONDING  STATES, 
and  we  deduce  from  our  equation  the  theorem  for  sub- 
stances to  which  van  der  Waals'  equation  applies,  that 
if  any  two  substances  have  temperatures  and  pressures 
bearing  the  same  ratios  to  their  critical  temperatures  and 
pressures,  then  their  volumes  will  bear  the  same  ratio  to 
their  critical  volumes.  This  is  the  theory  of  corre- 
sponding states,  and  may  be  further  extended  as  fol- 
lows :  If  we  call  the  ratio  of  the  vapor  tension  of  a 
saturated  vapor  to  the  critical  pressure 

-x: 

substituting  for  P  its  value  II/c,  the  equations  for  the 
reduced  vapor  tension  are 


n 


3*1- 

H=— ; 


from  which  we  conclude  that  II,  <£p  $3,  are  related  to 
$  by  purely  numerical  equations,  and  hence  two  vapors 
at  the  same  reduced  temperatures  have  the  same  re- 
duced pressures,  and  the  reduced  volumes  of  the  liquid 
and  saturated  vapor  are  the  same.  Now  all  our  quan- 
titative statements  regarding  the  critical  phenomena 
and  the  theory  of  corresponding  states  have  been 
expressly  limited  to  substances  following  van  der 


142  KINETIC   THEORY. 

Waals'  equation.  Experiment  alone  can  tell  whether 
the  theory  of  corresponding  states  is  more  than  an 
interesting  bit  of  mathematical  work.  That  it  has 
some  value  is  shown  by  so  simple  a  case  as  the  fol- 
lowing, which  we  owe  to  van  der  Waals  :  The  critical 
pressure  for  SO2  is  78.9  atmospheres,  and  the  critical 
temperature  428.4°  (absolute),  those  of  ether  36.9 
and  463°  respectively.  For  SO2  at  the  temperature 
412.9°  the  vapor  tension  is  60  atmospheres.  Then 


For  ether  at  the  same  reduced  pressure,  the  absolute 
pressure  is 

p=  n/c  =  .7605  x  36.9  =  28.4, 

and  according  to  Sajotschewski  this  pressure  corre- 
sponds to  an  absolute  temperature  of  445.8°,  whose 
reduced  value  would  be 

^=.963, 
463 

which  is  in  close  enough  agreement  with  the  value 
.964  for  SO2.  The  most  complete  test  has  been  made 
by  S.  Young  in  a  series  of  experiments  which  space 
does  not  permit  us  to  reproduce  entire.  He  made 
comparisons  of  quite  a  number  of  substances,  the 
names  and  critical  data  of  some  of  which  are  given  in 
the  following  table  : 


EQUATION   OF   VAN    DER   WAALS.  143 

CRITICAL  DATA. 


Substance. 

Formula. 

Mol.  Wt. 

r. 

/c 

*• 

Fluor-benzol, 

C6H5.F 

95-8 

559-55 

33>912 

2.822 

Chlor-benzol, 

C6H5.C1 

112.2 

633.00 

33,912 

2.731 

Brom-benzol, 

C6H5.Br 

156.6 

670.00 

33,912 

2.056 

lodo-benzol, 

C6H5I 

203.4 

721.00 

33,912 

i-7I3 

Benzol, 

C6H6 

77.84 

561.50 

36,395 

3-293 

Carbon  tetrachloride, 

CC14 

I53.45 

556.15 

34,180 

1.799 

Stannic  chloride, 

SnCl4 

259.3 

591  7° 

28,080 

1-347 

Ethei, 

(C2H5)20 

73.84 

467.40 

27,060 

3.801 

Methyl  alcohol, 

CH3.OH 

31.93 

513-00 

59,76o 

3-697 

Ethyl  alcohol, 

C2H5.OH 

45-90 

516.10 

47,850 

3-636 

Propyl  alcohol, 

C3H7.OH 

59.87 

536.70 

38,120 

3.634 

Acetic  acid, 

CH3.COaH 

59-86 

594.60 

43,400 

2.846 

In  this  table  pc  is  given  in  millimeters  of  mercury. 
The  values  for  the  critical  volumes  are  not  observed 
directly,  but  extrapolated  for  the  critical  temperature 
in  accordance  with  the  rule  of  Cailletet  and  Mathias 
that  the  arithmetic  mean  of  the  densities  of  the  liquid 
and  saturated  vapor  is  a  linear  function  of  the  tem- 
perature. 

The  following  tables  contain  some  of  the  results  of 
Young's  work,  embodied  in  a  comparison  between 
the  behavior  of  fluor-benzol  and  the  other  substances 
investigated.  The  subscript  o  will  refer  to  fluor-ben- 
zol. When  the  reduced  pressures  were  the  same, 
that  is,  n/II0  =  i,  the  ratios  of  the  other  reduced  co- 
ordinates were  found  to  vary  as  follows,  subscript  I 
referring  to  the  liquid  state  and  3  to  the  saturated 
vapor : 


144 


KINETIC   THEORY. 


Substance. 

from             to             from           to 

<Pi/<Poi 
from             to 

C6H5C1 

I.I309                          I.I37 

1.1246 

C6H5Br 

I.I976                          1.189 

1.  1802 

C6H5I 

1.2885                          I-282 

1.2772 

C6H6 

0.9890      1.0035                 0.946 

0.9439 

CC14 

0.9699      0.9939        0.969      I.OI7 

1.0173      1.0248 

SnCl4 

1.0679     I-°575             2.282 

1.2700      1.2910 

(C.2H5)20 

0.8431    0.8353             I-°35 

1.0319      1.0456 

CHgOH 

1.0127    0.9168      0.533    0-473 

0.4317     0.4172 

C2H5OH 

1.0494    0.9223      0.706    0.625 

0.6307     0.6058 

C3H7OH 

1.0997    O-9592      0.903    0.836 

0.8198     0.7937 

CH8COOH 

1.1278     1.0626      0.545    0.631 

0.6342 

When  the 

reduced  temperatures  wer 

e  the  same,  that 

is,  #/#0  = 

I,  the  ratios  of  the  other 

reduced  coordi- 

nates  were 

found  to  vary  as  follows  : 

Substance. 

from            to               from           to 

from              to 

C6H5C1 

0.998      .007               I«I39 

I.I245 

C6H5Br 

0.987        .007                    I.I94 

I.ISOO 

C6H5I 

0.985        .007                    1.286 

1.2773 

C6H6 

1.338        -073           0.842      0.940 

0.9479 

CC14 

1.460        .008          0.828      1.401 

1.0266 

SnCl4 

0.706     0.828           1.336      1.262 

1.2657      1.2913 

(C2H5)20 

0.705     0.798         i.ioi     1.025 

1.0284     l-°9%3 

CHgOH 

0.359    1.762       1.244    0.484 

0.409 

C2H5OH 

O.2OI       I.4II           2.076      0.648 

0.6044    0.5900 

C3H7OH 

0.099     1.124        2.678    0.866 

0.772 

CH3COOH 

0.476     1.280        0.894    0.631 

0.623 

In  these  tables  the  behavior  of  closely  allied  sub- 
stances shows  in  general  a  satisfactory  agreement  with 
the  theory  of  corresponding  states,  as  shown  both  by 
the  narrow  limits  between  which  the  values  lie,  and 
the  nearness  of  these  ratios  to  unity,  but  substances 


EQUATION   OF   VAN    DER   WAALS.  145 

of  widely  different  types,  such  as  alcohols  and  their 
derivatives,  show  marked  and  systematic  variations, 
hence  while  in  its  qualitative  form  this  theory  may 
apply  to  bodies  of  similar  constitution,  it  is  evident 
that  it  cannot  apply  without  further  explanation  or 
amplification  to  bodies  of  diverse  constitution. 

Another  method  of  discussion  was  also  suggested 
by  Mr.  Young,  which  has  been  carried  into  more 
detail  in  an  interesting  paper  on  the  critical  state  by 
Dieterici.1  In  the  equation 

RT 


substituting  for  b  its  value  vj$  we  have 

v       RT 
'      3V^3+-*> 

RT 


Now  RTJpc  is  the  volume  which  an  ideal  gas  would 
have  at  the  critical  pressure  and  temperature,  which 
we  may  call  the  ideal  critical  volume,  and  indicate  by 
vk.  Then  for  a  gas  following  van  der  Waals'  equa- 
tion 


or  the  ideal  critical  volume  is  |  of  the  actual  critical 
volume.     A  table  is  given  collecting  the  results  of  the 

1  Wied.  Ann.,  69,  pp.  685-705,  1899. 

10 


146  KINETIC   THEORY. 

determinations  of  Ramsey  and  Young  which  bear 
upon  this  point.  The  ratio  is  in  no  case  |,  that  is 
2.667,  but  for  the  benzol  derivatives  and  some  others 
has  values  very  nearly  3.75,  for  a  considerable  list  of 
esters  the  values  lie  close  to  3.8,  for  ethyl  and  pro- 
pyl  alcohol,  4.02,  while  the  values  for  methyl  alcohol, 
4.52,  and  acetic  acid,  5.00,  are  entirely  different. 
Something  the  same  range  of  values  is  found  by  col- 
lating the  results  of  other  workers  for  different  sub- 
stances, as  found  in  original  papers  and  Landolt  and 
Bernstein's  tables.  The  direct  determination  of  the 
critical  volume  is  a  matter  of  considerable  difficulty, 
and  the  tendency  of  the  systematic  errors  to  be  ex- 
pected is  to  give  its  value  too  small,  and  hence  the 
ratio  of  the  ideal  to  the  actual  critical  volume  too  large. 
Taking  these  facts  into  consideration,  the  declaration 
of  Mr.  Young  seems  to  be  substantiated  that  for  all 
substances  which  can  attain  the  critical  state  without 
chemical  change,  this  ratio  is  very  nearly  the  same, 
being  not  far  from  3.7.  As  has  been  shown  the  equa- 
tion 

(33) 

leads  to  the  value  2.667.  Other  formulae  have  been 
proposed  which  suggest  the  possibility  of  closer  ap- 
proximation by  the  increased  number  of  constants 
available,  as 


EQUATION    OF   VAN    DER   WAALS. 


1  47 


where  P  may  be  of  the  form  ajv2,  but  these,  on  inves- 
tigation lead  to  values  of  our  ratio 


of  which  the  results  from  van  der  Waals'  original  form 
are  a  special  case.  Dieterici  proposes  the  following 
entirely  empirical  formula,  which  is  of  a  form  not  un- 
like that  of  van  der  Waals  ;  if  we  put  the  "  molecular 
pressure"  equal  to  ajv*  our  equation  becomes 


(42) 


Multiplying  up  and  arranging  according  to  the  powers 
of  v 

a         ab 


We  may  introduce  the  new  variable  x=  ifi  and  write 
our  equation  symbolically 

f(x)  =  x*  —  ax5  +  ft  A?  —  7  =  o. 

This  equation  is  of  the  eighth  degree,  but  can  accord- 
ing to  the  theory  of  equations  have  at  most  three  real 
positive  roots.  It  has  also  one  negative  root  and  four 
imaginary  ones.  At  the  critical  point  the  three  posi- 
tive roots  coincide  and  we  have 


f(x) 
f(x) 
f'(x) 


—  ax5 
7  -  5  our 


—  7  =  0, 


=  o. 


148  KINETIC    THEORY. 

Solving  these  three  equations  for  a,  /3  and  7,  we  nave 
a  =  4**,          /3  =  4*5,          7  =  .r8, 

or  introducing  the  values  of  a,  ft  and  7,  and  calling 
the    critical    volume,    pressure,    and    temperature    as 

before  v#  pc,  Te, 

RT 


ab 

-p 

whose  solutions  are 


or 

v  = 


equations  which  are  entirely  analogous  to,  though  of 
slightly  different  form  from  those  of  van  der  Waals  for 
the  critical  constants.  Substituting  the  values  of  a  and 
b  in 


EQUATION   OF  VAN   DER  WAALS.  149 

we  have 


—  -  a  R- 

"  T°    ' 


or  introducing  the  reduced  volume,  pressure  and  tem- 
perature 

(43) 

From  the  foregoing  it  is  apparent  that  this  equation 
has  the  same  general  properties  as  the  form  due  to 
van  der  Waals,  including  the  theory  of  corresponding 
states,  which  depends  upon  the  fact  that  the  equation 
relating  the  reduced  pressure,  volume  and  temperature 
is  purely  numerical.  Still  further,  in  the  equation 

RT 

>+:-,-'  -4». 

putting  for  b  its  value  vj^.  and  for  RTJpc  as  before  vk 


JT-3-75. 


This  value  corresponds  very  closely  to  that  found  by 
Mr.  Young  from  his  experiments. 

Dieterici  even  goes  farther,  and  deduces  from  theo- 
retical considerations  a  formula  connecting  the  pres- 
sure, volume  and  temperature  of  the  form l 

RT          • 

(44)  /==__.,     TVf 

lSee  Chapter  VII.,  p.  171. 


ISO  KINETIC   THEORY. 

and  since  for  the  critical  point  dpjdv—o  an 

he  is  able  to  find  the  value  of  the  ratio  of  the  ideal  to 

the  actual  critical  volume,  which  proves  to  be 

*-  3.695. 

Berthelot  l  has  shown  that  the  actual  isothermal  for 
carbon  dioxide  through  its  critical  point  agrees  almost 
exactly  with  that  given  by  van  der  Waals'  equation, 
taking  the  constants  of  the  latter  from  the  coordi- 
nates of  the  critical  point,  for  pressures  greater  than 
the  critical  pressure,  that  is,  in  the  region  near  the 
liquid  state,  while  for  smaller  pressures  and  the  vapor 
state  the  actual  isothermal  follows  very  closely  that 
givea  by  a  special  form  of  Clausius'  equation, 


or  in  reduced  coordinates 


while  an  equation  can  be  found  giving  a  curve  very 
nearly  agreeing  with  the  actual  isothermal  by  writing 
for  a/v2, 

a 


v*  +  2lvb  -f 
which  gives  a  reduced  equation  of  the  form 


1  Comptes  Rendus,  130,  pp.  69  and  115,  1900. 


EQUATION   OF   VAN   DER  WAALS.  151 

He  finds  further  that  van  der  Waals'  equation  gives 
for  liquids  isothermals  of  the  same  general  shape  as 
the  actual  ones,  but  very  differently  situated.  At- 
tempting to  introduce  some  form  of  correction  which 
should  make  the  two  isothermals  coincide,  he  finds 
this  is  best  done  by  regarding  b  as  a  function  of  the 
temperature.  The  empirical  form 


obtained  from  the  study  of  the  isothermals  of  liquid 
carbon  dioxide,  which  leads  to  the  reduced  equation 


gives  isothermals  almost  identical  with  those  for  car- 
bon disulphide  and  ethyl  chloride  at  o°  C.  He  con- 
cludes from  this  that  the  equation  of  van  der  Waals, 
if  the  volume  b  be  regarded  as  dependent  upon  the 
temperature,  represents  satisfactorily  the  behavior  of 
normal  liquids.  The  objections  which  have  been 
raised  against  it  are  due  largely,  he  believes,  to  the 
fact  that  our  experimental  study  has  naturally  been 
more  of  vapors  under  their  relatively  smaller  pres- 
sures than  of  liquids  with  their  relatively  larger  pres- 
sures. 


CHAPTER   VII. 
VAPORIZATION. 

THE  study  of  the  motions  of  the  molecules  of  a 
liquid  may  be  approached  in  three  not  entirely 
dissimilar  ways.  Equations  of  the  type  of  that  of 
van  der  Waals  appear  to  apply  to  the  liquid  as  well 
as  to  the  gaseous  state,  both  because  of  the  resem- 
blance between  the  forms  of  their  isothermal  curves 
and  those  found  from  experiments,  and  because  the 
conditions  underlying  the  assumptions  which  lead  to 
these  equations  are  even  more  characteristic  of  liquids 
than  of  gases.  A  second  method  of  approaching  the 
study  of  the  liquid  state  is  by  giving  attention  to  the 
phenomena  of  change  of  state,  while  a  third  consists 
in  the  direct  attack  upon  the  problem  of  the  motion 
of  molecules,  the  mean  distance  between  which  is  of 
the  order  of  their  dimensions. 

The  first  method  has  been  pursued  with  some  suc- 
cess by  Traube l  of  the  Technische  Hochschule,  Ber- 
lin. He  distinguishes  carefully  between  the  volume 
of  the  substance  of  the  atoms  which  make  up  the 
molecule,  the  volume  of  the  molecule,  which  may  be 
due  not  only  to  the  bulk  of  the  atoms,  but  also  to 
their  arrangement,  and  the  "co-volume,"  the  space 
not  occupied  by  the  molecules,  in  which  the  individual 

1  Wied.  Ann.,  61,  pp.  380-400,  1897. 
152 


VAPORIZATION.  153 

molecule  is  free  to  move.  This  co- volume  is  the  v  —  b 
of  the  generalized  formula 

(45)  (p  +  P)(v.-b)  =  RT. 

In  the  case  of  a  liquid  the  external  pressure  /  is  so 
small  in  comparison  with  the  "molecular  pressure" 
that  it  may  be  disregarded,  and  calling  the  co-volume 
<£  the  equation  becomes 

(46)  P*  =  RT, 

that  is,  the  product  of  the  molecular  pressure  by  the 
co-volume  of  a  liquid  is  proportional  to  the  absolute 
pressure.  While  this  result  is  interesting,  it  can  only 
be  verified  by  experiments  which  shall  give  us  values 
of  both  P  and  O  for  a  variety  of  substances  in  the 
liquid  states.  The  volume  of  a  gram  molecule  of  a 
gas  at  o°  C.  and  76  cm.  pressure  is  22,380  c.c.,  a  figure 
which  may  also  be  regarded  as  representing  the  co- 
volume.  For  normal  liquids,  that  is  those  in  which 
the  molecules  contain  the  number  of  atoms  called  for 
by  their  formulae,  Traube  finds,  by  deriving  the  volume 
of  the  molecules  from  determinations  of  refractive  in- 
dices, that  for  the  same  pressure  and  temperature  the 
co-volume  of  a  gram-molecule  is  very  nearly  24.5  c  c., 
and  hence  that  the  molecular  pressure  for  such  liquids 
is  22,380/24.5  =  913  atmospheres.  This  value  is  not 
very  different  from  that  found  by  Nernst  for  carbon 
dioxide  from  the  formula  P  =  a/v2  which  is  970  at- 
mospheres.1 The  complete  verification  of  this  form 

1  Nernst,  "Theoretical  Chemistry,"  Trans,  by  Palmer,  p.  196. 


154  KINETIC   THEORY. 

of  the  theory  would  seem  to  depend  upon  experi- 
mental determinations  of  P.  It  may  be  that  experi- 
ments upon  the  thermal  expansion  and  coefficients  of 
compressibility  of  liquids  will  furnish  the  necessary  data. 
Equilibrium  Between  Liquid  and  Vapor. — The  second 
method  is  well  summarized  by  Dieterici l  whose  treat- 
ment we  shall  follow.  Any  demonstration  which  may 
be  given  for  the  purpose  of  establishing  the  Maxwell 
distribution  of  the  velocities  of  the  molecules  of  a  gas 
depends  fundamentally  upon  the  assumption  that  the 
number  of  the  molecules  is  exceedingly  great,  and  that 
a  knowledge  of  the  speed  and  direction  of  one  mole- 
cule, or  of  any  number,  gives  no  clue  as  to  the  speed 
or  direction  of  any  other  molecule.  Hence  we  can 
believe  that  this  distribution  is  just  as  applicable  to 
and  just  as  probable  for  the  motion  of  the  molecules 
of  liquids  as  of  gases.2  The  attractive  forces  between 
the  molecules,  as  we  have  indicated  in  the  last  chap- 
ter, cannot  particularly  influence  the  motion  of  the 
molecules  except  near  the  surface.  Let  us  suppose 
then  that  we  have  a  substance  present  in  both  the 
liquid  and  vapor  form,  with  the  surface  of  separation 
horizontal.  We  may  suppose  further,  for  the  sake  of 
simplicity,  that  each  follows  the  laws  of  ideal  gases, 
except  near  their  surfaces,  and  that  each  exhibits  Max- 
well's distribution  of  velocities.  While  we  may  speak 
of  the  plane  of  separation  of  the  two  phases,  they  are 
really  separated  not  by  a  plane,  but  by  a  non-homo- 

>  Wied.  Ann.,  66,  pp.  826-858,  1898. 
2Rayleigh,  Phil.  Mag.  (5),  49,  p.  1900. 


VAPORIZATION.  155 

geneous  layer.  If  the  line  O  represents  the  position 
of  the  mathematical  surface  of  separation,  then  below 
it  we  may  consider  a  plane  to  be  passed,  which  we 
represent  by  B,  at  a  sufficient  distance  so  that  all 
the  liquid  below  B  can  be  regarded  as  completely 
homogeneous,  while  in  the 
space  between  B  and  0  a 

molecule  as  it  approaches  0    o 

is   subject    to    stronger   and 
stronger    forces    tending    to  Fi    u 

draw    it    back    toward    the 

body  of  the  liquid.  Similarly  we  may  pass  a  plane  A 
above  0,  at  such  a  distance  that  above  A  the  vapor 
will  be  homogeneous,  while  only  between  0  and  A 
will  it  be  non-homogeneous,  being  denser  near  0  on 
account  of  the  attraction  of  the  molecules  of  the  liquid. 
Only  the  vertical  components  of  the  motions  of  the 
molecules  tend  to  carry  them  from  one  of  the  regions 
into  the  other,  hence  our  discussion  is  restricted  to  the 
vertical  components  of  the  velocities,  which  we  shall 
designate  by  the  letter  u.  We  shall  indicate  quantities 
referring  to  the  region  of  vapor  generally  by  the  sub- 
script a  and  those  referring  to  the  liquid  state  by  the 
subscript  b,  the  constants  of  Maxwell's  formula  being 
a  and  /3  respectively.  Then  the  number  of  molecules 
in  the  unit  volume  of  the  liquid  having  the  vertical 
components  of  their  velocities  between  u  and  u  +  du  is 


156  KINETIC   THEORY. 

and  in  the  unit  volume  of  the  vapor  similarly 

-^i^dti. 

If  as  a  first  approximation  we  assume  that  both  phases 
of  the  substance  follow  the  laws  of  ideal  gases,  the 
number  of  such  molecules  striking  a  unit  area  of  the 
surface  B  in  one  second  will  be  the  first  of  these  num- 
bers multiplied  by  the  speed  uy  that  is, 

•*7  5_ 

e  P*  udu. 


7T 


Now  of  all  the  molecules  which  strike  the  surface  B 
from  below,  a  part  only  go  some  distance  into  the  non- 
homogeneous  layer,  and  then  return  on  account  of  the 
strong  unbalanced  force  they  meet  there,  while  some 
go  clear  through  beyond  the  plane  A  into  the  region 
of  vapor.  In  general,  disregarding  for  the  present  the 
effects  of  collisions,  molecules  having  the  vertical  com- 
ponents of  their  velocities  greater  than  a  certain  mini- 
mum value  which  we  may  call  s  will  be  capable  of 
passing  up  entirely  through  the  non-homogeneous 
layer  into  the  region  of  vapor ;  these  we  may  call 
briefly  the  "  capable "  molecules ;  while  molecules 
having  the  vertical  component  less  than  this  amount 
will  penetrate  to  a  greater  or  less  distance  into  the 
space  between  B  and  A,  and  then  return  to  the  liquid. 
Then  the  total  number  of  molecules  which  will  pass 
in  one  second  through  a  unit  area  of  the  surface  B 
into  the  region  of  vapor  will  be  the  sum  total  of  all 


VAPORIZATION.  157 

the  molecules  the  vertical  component  of  whose  veloc- 
ities is  greater  than  s,  or  calling  this  number  nt, 


J°°     n       _— 
—  b-=e  ^ndu. 
.     PVir 


This  expression  is  readily  integrated  by  putting  -=  =  x, 
giving 


The  same  reasoning  shows  that  the  number  of  mole- 
cules of  the  vapor  having  the  vertical  components  of 
their  velocities  between  u  and  u  +  du  which  strike  unit 
area  of  the  surface  A  from  above  in  one  second  is 


Now,  barring  mutual  collisions,  there  is  nothing  to 
prevent  any  molecules  passing  down  through  A  from 
passing  completely  into  the  liquid,  or  rather,  on  ac- 
count of  the  attraction  of  the  liquid  they  must  inevit- 
ably pass  down  into  it,  hence  the  total  number  of 
molecules  passing  down  through  unit  area  of  A  in  one 
second  and  entering  the  liquid  is 

fna         *  n  a. 

-a-=e  °-*udu  =  —a-=. 
a~l/7T  21/7T 


1  58  KINETIC   THEORY. 

and  when  a  state  of  equilibrium  is  attained  this  must 
be  the  same  as  nt,  the  number  passing  from  the  liquid 
to  the  vapor  or 


2T/ 


7T 


Hence  the  condition  of  equilibrium  that  equal  num- 
bers of  molecules  pass  into  and  out  of  the  liquid  in 
the  same  time  gives  us  the  relation 


We  shall  next  consider  the  amount  of  energy  which 
the  molecules  carry  with  them  in  their  passage  into 
and  out  of  the  liquid.  We  shall  assume  that  the  mass 
of  the  molecules  is  the  same  in  both  states,  that  is, 
that  there  is  no  dissociation  or  association  of  the  mole- 
cules accompanying  the  change  of  state.  Each  mole- 
cule having  the  vertical  component  of  its  velocity 
equal  to  u  has  associated  with  that  component*  the 
energy  ^miP.  Then  the  total  energy  of  the  molecules 
having  components  between  u  and  u  -f  du  which  pass 
up  through  a  unit  area  of  B  in  a  second  is 


and  the  molecules  which  pass  through  this  unit  area 
in  a  second  into  the  region  of  vapor  carry  with  them 
a  total  energy 


n        _^  nm 

Vudu  =  —±-=       e 


VAPORIZATION.  159 

This  integral  is  easily  evaluated  by  the  same  substitu- 
tion as  before  and  by  application  of  formula  (8)  of  p. 

27,  giving 


n.mj&  r 
—j=  \ 

2  1/7T  J5/0 


which  reduces,  after  writing  ng  for  its  value,  to  the 
form 


This  then  is  the  amount  of  energy  taken  out  of  the 
liquid  by  the  ns  molecules  leaving  it.  Now  consider 
a  molecule  the  vertical  component  of  whose  velocity 
is  exactly  s  ;  such  a  molecule  would  have  associated 
with  this  motion  the  energy  \ms^  ;  having  exactly  the 
speed  s  it  would  just  barely  penetrate  through  A  to 
the  homogeneous  region  arriving  there  without  motion 
and  without  energy.  The  energy  \ms*  which  it  has 
lost,  then  represents  exactly  the  work  which  the 
molecule  must  do  in  overcoming  the  attraction  of  the 
mass  of  liquid  and  penetrating  through  the  non-homo- 
geneous layers  between  B  and  A.  The  remainder, 
\nm^  then  represents  the  energy  which  the  capable 
molecules  bring  with  them  into  the  vapor  region. 
Similarly  the  molecules  passing  down  from  the  vapor 
into  the  liquid  carry  with  them  the  energy 


160  KINETIC   THEORY. 

n  m      /»ao  -^    . 
— ^—=  I    e  °?uz  du 
2ay7r  Jo 

which  is  easily  shown  to  be  equal  to 


and  the  condition  of  equilibrium  that  equal  amounts 
of  energy  shall  be  associated  with  the  molecules  pass- 
ing into  and  out  of  the  vapor  gives  us  the  condition 


That  is,  the  most  probable  speeds  of  the  molecules  of 
the  liquid  and  vapor  states,  and  hence  their  average 
speed,  and  their  mean  kinetic  energy  of  translation 
are  the  same.  Knowing  that  their  temperatures  must 
be  the  same  this  is  a  result  that  might  reasonably 
have  been  expected.  Introducing  this  into  the  rela- 
tion between  na  and  nb  we  have 


that  is,  since  na  and  nh  are  proportional  to  the  densi- 
ties of  the  vapor  and  liquid  states  respectively,  the  ratio 
of  these  densities  is  equal  to  e~**la2  where  s  is  the  speed 
necessary  to  penetrate  the  non-homogeneous  layer, 
and  a  the  most  probable  speed  of  the  molecules. 

We  have  next  to  consider  the  momenta  associated 
with  the  molecules  of  the  liquid  and  vapor  phases  of 
the  substance.  This  momentum  carried  by  the  mole- 
cules is  proportional  to  the  internal  pressures.  The 


VAPORIZATION.  l6l 

sum  of  all  the  momenta  brought  from  below  to  a  unit 
area  of  the  surface  B  in  one  second  is 


f 


but  of  this  amount  only  that  associated  with  mole- 
cules having  u  at  least  as  great  as  s  can  ever  pass 
through  the  surface  A  to  the  region  of  vapor,  and  this 
momentum  is 


which  may  be  reduced  by  the  methods  previously 
employed  to  the  form 


nhm\  s   _l 
2  V  TrLP 


f06  1 

e       +        e-+dx  I 
Js/p  J 


of  which  the  integration  of  the  last  term  is  affected 
only  by  development  in  series  or  by  the  use  of  tables. 
But  not  all  this  momentum  which  the  capable  mole- 
cules carry  with  them  through  the  surface  B  is  also 
carried  with  them  through  the  surface  A,  but  only 
that  which  is  associated  with  the  excess  of  this  com- 
ponent of  the  velocity  over  the  critical  speed  s.  This 
is  conditioned  by  the  energy  relation  for  any  molecule 
which  passes  through  the  non-homogeneous  layer, 


consequently  the  total    momentum    passing    upward 
through  unit  area  of  the  surface  A  is 
ii 


1 62  KINETIC   THEORY. 


which  is  again  an  exceedingly  difficult  form  to  inte- 
grate. We  may  however  attempt  a  different  method 
of  treatment.  If  the  energy  of  all  the  molecules  asso- 
ciated with  the  -f  u  motion  could  suddenly  be  de- 
creased by  the  uniform  amount  \m&,  then  immediately 
after  this  change  the  law  of  distribution  of  u  would  be 


n,      -          7 
ft*  du. 


Now  the  effect  of  the  forces  acting  in  the  non-homo- 
geneous layer  is  exactly  the  same  as  if  they  were  able 
to  impress  upon  each  of  the  capable  molecules  such  a 
negative  velocity  as  should  serve  to  decrease  its  energy 
by  \ms2,  and  hence  to  produce  in  them  just  such  a 
distribution  of  velocities  as  is  indicated  by  the  above 
form  of  Maxwell's  law,  where  u  will  refer  to  the  actual 
component  velocities  possessed  by  the  molecules  on 
reaching  the  surface  A.  Then  the  sum  of  the  momenta 
which  the  capable  molecules  will  carry  through  unit 
area  of  A  in  one  second  is  l 


_  . 

ftf,       _^±A2  njne    f*2  r*    _«! 

mu—b=.e      ^    udu=    *      ,-     \     e    ** 
j3l/7T  PV*      Jo 


j3l/ 

which  may  be  shown  by  the  methods  of  p.   27  to 
be  equal  to 

1  This  same  method  might  have  been  applied  to  the  study  of  the 
passage  of  masses  and  energy  through  the  surface  A,  leading  to  the 
same  results  as  those  we  have  obtained. 


VAPORIZATION.  163 


*•    */9. 


47r  21/7T 

The  sum  of  the  momenta  carried  down  through  unit 
area  of  the  surface  A  is  similarly 


X00          n 
mu  — =  e 


=  -^=J    e    *itdu 
=  >V^  =      na 

47r  21/7T 

Hence  the  two  relations  which  we  have  already  de- 
duced that 

«j8     -  4        na 


and  that 
and  hence 


Mg  —  "~~  , —  £         —        > —  > 

21/7T  21/7T 


satisfy  not  only  the  conditions  that  in  a  state  of  equi- 
librium the  masses  passing  through  the  surface  A  in 
the  two  directions,  and  the  kinetic  energies  associated 
with  them  are  equal,  but  also  that  the  momenta  asso- 
ciated with  them,  and  hence  the  external  pressures  are 
equal. 

A  careful  review  of  the  three  problems  we  have 
just  studied  shows  that  the  three  equations  which  we 


164  KINETIC    THEORY. 

have  obtained  as  indicating  equilibrium  of  mass  or 
number  of  molecules,  of  energy,  and  of  momenta 
are,  omitting  common  constant  factors 


n  a.  =  n, 


The  inevitable  conclusion  from  these  different  equations 
is  that  a  =  /3,  and  that  the  meaning  of  this  is,  not  simply 
that  some  energy  associated  with  the  motions  of  the 
molecules  in  the  liquid  state  is  equal  to  a  correspond- 
ing energy  in  the  gaseous  state,  but  that  the  mean 
kinetic  energy  of  translation  is  the  same  in  both  states, 
but  not  necessarily  the  total  energy  of  the  molecules. 
This  condition  makes  the  condition  of  thermal  equi- 
librium between  liquid  and  vapor  definite  and  the  same 
as  that  between  two  gases.  This  conclusion  may 
otherwise  be  stated  that  the  mean  kinetic  energy  of 
translation  of  the  molecules  is  the  measure  of  the 
temperature  in  liquids  as  in  gases,  or 

\Nrn~?  =RT. 

Review  of  Assumptions.  —  In  the  foregoing  discus- 
sion we  have  assumed  that  the  Maxwell  distribution 
of  velocities  holds  equally  well  for  the  molecules  of 
liquids  and  gases,  that  the  mass  of  the  molecules 
is  the  same  in  both  states,  and  that  the  volumes  of 


VAPORIZATION.  165 

the  molecules  can  be  entirely  disregarded.  Of  these 
assumptions  the  first  we  believe  to  be  valid,  and  the 
second  we  regard  as  consistent  with  the  facts  in  a 
great  many  cases  ;  the  discussion  of  the  consequences 
of  any  deviation  from  this  assumption  can  well  be 
deferred.  The  third  assumption  has  two  immediate 
consequences ;  we  have  disregarded  the  effect  of  the 
volume  occupied  by  the  molecules  upon  the  number 
of  molecules  which  will  pass  through  any  area  in  a 
given  time,  and  we  have  entirely  ignored  the  possibility 
of  mutual  collisions.  If  we  apply  to  this  case  the 
results  of  the  discussion  in  Chapter  III.  of  the  effect 
of  the  volume  of  the  molecules  when  this  is  small, 
but  not  negligibly  small  in  comparison  with  the  vol- 
ume of  the  gas  or  liquid,  if  the  volume  occupied  by  a 
number  N  of  the  molecules  of  the  substance  in  the 
vapor  state  is  va,  and  in  the  liquid  state,  vb>  we  have 
to  consider  as  the  number  of  molecules  per  unit  vol- 
ume, not  the  number  per  unit  total  volume,  Njva  or 
Njvb  but  the  number  per  unit  of  co-volume ',  that  is 

N 


N 

^      i)  — 
and  hence 


n,  v  —  b  y 

o  a,  a 

or 

v  -b          *. 


1  66  KINETIC   THEORY. 

and  we  have,  not  the  ratio  of  the  densities,  or  of  the 
specific  volumes  in  the  two  states,  but  of  the  co-volumes 
determining  the  relation  between  s  and  the  actual 
speeds  of  the  molecules. 

In  ignoring  the  possibility  of  collisions  between  the 
molecules  we  have  assumed  that  the  particular  mole- 
cules whose  conditions  we  considered  at  the  surface  B 
passed  right  up  through  the  non-homogeneous  layer 
into  the  region  of  vapor  above  A  ;  but  in  reality  all 
this  region  between  A  and  B  is  filled  with  either  the 
liquid  or  the  vapor  in  very  dense  form,  so  that  almost 
never  could  a  molecule  pass  up  directly,  but  it  is  much 
more  likely  to  strike  other  molecules,  and  by  the  im- 
pact transmit  upward  its  energy  and  momentum,  and 
keep  intact  the  number  of  molecules  traveling  upward. 
This  will  not  affect  our  conclusions  with  regard  to  the 
equilibrium  between  the  number  of  molecules  passing 
through  the  layer  in  both  directions,  nor  with  regard 
to  the  equilibrium  in  the  state  of  momentum,  or  of 
pressure,  but  will  compel  us  to  rediscuss  the  problem 
of  the  energy  relations. 

The  energy  of  the  capable  molecules  passing  up 
through  unit  area  of  B  in  one  second  was  shown  to  be 


and  our  interpretation  was  perfectly  general  ;  that 
^ms2,  being  the  energy  of  a  molecule  just  exactly 
capable  of  penetrating  the  non-homogeneous  layer 
was  the  work  that  must  be  done  by  each  molecule 
against  the  cohesive  forces  of  the  liquid,  while  %ntmf& 


VAPORIZATION.  1  67 

was  the  energy  associated  with  the  vertical  component 
of  the  velocities  of  the  ns  molecules  after  penetrating 
this  layer.  An  obvious  interpretation  is  that  ±nms2 
represents  the  latent  heat  of  vaporization,  measured  in 
dynamical  units,  but  attempts  to  verify  this  suggestion 
by  reference  to  numerical  data  lead  to  such  inconsis- 
tencies and  confusion  that  we  prefer  to  seek  for  a  dif- 
ferent interpretation,  rather  than  fill  our  pages  with  a 
statement  of  what  cannot  be.  So  far  we  have  drawn 
all  conclusions  from  considerations  of  equilibrium,  but 
the  value  of  the  latent  heat  of  vaporization  is  to  be 
found  not  by  a  study  of  conditions  of  equilibrium,  but 
of  the  amount  of  energy  required  to  change  a  definite 
quantity  of  the  substance  from  the  liquid  to  the  vapor 
state.  Disregarding  for  the  present  changes  in  the 
energy  of  the  internal  motion  of  the  molecules,  which 
might  give  a  corrective  term,  this  energy  is  equal  to 
the  work  which  must  be  done  against  all  the  forces, 
whether  cohesive  or  of  external  pressure,  in  the  expan- 
sion of  the  liquid  to  the  gas.  Now  we  have  shown  in 
Chapter  III.,  p.  69  that  the  total  of  these  forces  is 

RT 


hence  the  work  in  a  small  expansion  dv  is 
RT 


j 

v  —  b 

and  the  total  work  in  expanding  from  the  liquid  to  the 
vapor  state  is 


RT 

'vb 


r»     dv 
.  ^irif 


168  KINETIC    THEORY. 

which,  if  we  assume  that  b  is  the  same  for  both  the 
liquid  and  the  vapor  states,  becomes 


If  we  substitute  for  (va  —  b)j(vb  —  b)  its  value  e8^^  and 
remember  that 


and  that 

?  =  f/32, 

this  expression  reduces  to  the  form 


which  is  the  same  as  the  value  we  have  just  obtained 
from  conditions  of  equilibrium.  But  in  considering 
the  transmission  of  energy  through  the  non-homo- 
geneous layer,  we  have  to  take  into  account  still 
another  effect  of  the  volume  of  the  molecules.  The 
intermolecular  forces  which  have  their  effect  in  the 
non-homogeneous  region  act  not  upon  the  energy  of 
the  molecules,  but  simply  on  the  molecules  them- 
selves, the  carriers  of  the  energy ;  consequently,  while 
any  molecule  is  progressing  through  this  layer  upward, 
it  is  doing  work  against  these  forces,  but  whenever  it 
comes  into  collision  with  another  molecule  it  transmits 
its  energy  a  certain  distance,  in  the  case  of  a  central 
collision  just  equal  to  the  diameter  of  the  molecule, 
without  loss  because  unaffected  by  these  intermolec- 
ular forces.  The  expression  which  we  have  just 


VAPORIZATION.  169 

deduced  represents  the  loss  of  kinetic  energy  as  the 
energy  is  transmitted  through  this  non-homogeneous 
layer.  But,  in  the  process  of  vaporization  of  a  liquid 
molecules  actually  pass  out  of  the  liquid  region  into 
the  vapor  region,  while  the  non-homogeneous  layer, 
even  if  not  composed  of  the  same  identical  molecules, 
remains  intact.  Hence  we  may  regard  the  molecules 
as  actually  passing  from  the  inner  homogeneous  region 
to  the  outer  homogeneous  region,  and  hence  doing  an 
amount  of  work  which  is  greater  than  that  which  we 
have  just  found  in  the  ratio  of  vjv  —  b.  The  latent 
heat  will  then  be  l 


va-b 

The   generalized   form   of  van   der  Waals'   equation 
applying  to  the  two  phases  is 

(P  +  ^)K  -*)-(/+  PM*.  -  *)  =  \Nm?  -RT, 
where  Pa  and  Pb  are  the  molecular  pressures  within 
the  vapor  and  the  liquid  respectively.     Substituting 
from  these  equations  we  have 


or 

(48)  L  =  RT  log  |±^  +  (/>  -  PJ6. 

1  Milner,  Phil.  Mag.  (5),  43,  pp.  291-304,  1897. 


1  70  KINETIC   THEORY. 

Since  this  expression  contains  only  the  work  which  is 
done  against  the  forces  acting  in  the  non-homogeneous 
layer,  and  not  the  work  done  against  the  external 
pressure,  in  pushing  back  the  containing  wall,  as  each 
part  of  the  gas,  fully  expanded,  comes  out  of  the  non- 
homogeneous  layer,  we  must  add  to  it  the  work  done 
against  the  external  pressure, 


giving 

L  =  R  T  log 

& 


We  have  already  shown  that  the  first  term  of  this 
result  for  the  latent  heat  of  vaporization  is  the  same 
as  the  value  we  have  previously  found  for  the  loss  of 
the  kinetic  energy  of  the  molecules  in  passing  through 
the  non-homogeneous  layer.  The  second  term  is  the 
product  of  the  molecular  pressure  in  the  liquid  state 
by  the  volume  of  the  liquid,  and  may  be  called  the 
potential  energy  of  the  liquid  film,  or  non-homoge- 
neous layer.  It  is  equal  to  the  work  which  would  be 
done  in  displacing  the  film  by  an  amount  equal  to  the 
volume  of  the  liquid,  against  the  molecular  pressure 
which  holds  it  extended.  This  is  what  actually  occurs 
in  the  vaporization  of  the  liquid.  The  third  term  is  a 
similar  expression  for  the  vapor  state,  but  of  much 
smaller  amount  ;  the  difference  between  the  two  poten- 
tial energies  represents  then  the  amount  of  work  which 
must  be  done  in  vaporizing  a  body  of  liquid,  in  addi- 
tion to  that  necessary  to  replace  the  loss  of  kinetic 


VAPORIZATION.  I/I 

energy  of  the  molecules  as  they  pass  through  the 
surface  layer. 

The  method  of  this  chapter  may  be  applied  to  the 
deduction  of  the  equation  of  the  vapor  or  of  the  liquid. 
The  pressure  exerted  upon  the  walls  of  the  containing 
vessel  by  a  vapor  or  liquid  depends  upon  the  sum  of 
the  impulses  of  the  molecules  that  actually  strike  the 
wall.  To  reach  the  wall  the  molecules  have  to  pass 
through  a  non-homogeneous  layer  in  which  forces  act 
which  tend  to  retard  the  approach  of  the  molecules 
to  the  wall.  The  expression  for  the  pressure  may 
then  be  written 

*       RT  e~£ 
p  =  ~^~be      ' 

where  s  is  the  speed  which  a  molecule  must  have  in 
order  to  just  penetrate  to  the  wall.  This  may  also  be 
written 

_  J?ZL  -5? 

*  ~  v  —  b 

where  A  is  an  amount  of  energy  proportional  to  the 
work  of  a  molecule  in  overcoming  the  cohesive  forces 
in  reaching  the  wall.  If  we  assume  that  this  quantity 
A  is  proportional  to  the  density,  we  may  write 

A-Z. 

V 

where  a  is  constant,  and 

RT        « 

(44)  /  =  -  —.*  **• 


172 


KINETIC   THEORY. 


The  condition  that  for  the  critical  state  the  first  and 
second  derivatives  of  /  by  v  vanish  gives  us 


a 


VAPORIZATION.  1/3 


which  is  almost  exactly  the  mean  value  found  by 
Young,  while  the  equation  between  the  reduced 
coordinates  is 

~ 


(49) 


If,  however,  in  the  equation 

^r    —  «. 

(44)  *-i^r&* 

we  assume  that  ajRTv  is  small,  we  may  write  for 
e-a\RTv  the  ^rst  two  terms  of  its  development,  I  —  a[RTv, 
giving 

RT  a 


or  disregarding  ^  in  the  last  term  as  small, 

RT        a 


which  is  van  der  Waals'  equation,  so  that  at  fairly 
high  temperatures  or  for  fairly  large  volumes  the 
equations  are  practically  identical,  a  and  b  having  the 
same  meaning,  though  they  give  entirely  different 
values  of  the  critical  constants. 


CHAPTER  VIII. 
MOLECULES   WITHIN   A   LIQUID. 

IN  Chapters  VI.  and  VII.  we  discussed  formulae  of 
the  form 
(45)  (f  +  Pyv-b)-RT, 

of  which  van  der  Waals'  equation  was  a  special  case, 
and  showed  that  they  could  be  applied  to  the  study 
of  liquids,  and  to  the  phenomena  of  vaporization,  as 
well  as  to  gases. 

That  the  special  equations  like  those  of  van  der 
Waals,  Clausius,  and  others  could  completely  describe 
the  behavior  of  liquids  was  not  to  be  expected  because 
they  were  deduced  on  the  assumption  that  the  free 
paths  of  the  molecules,  while  not  necessarily  infinitely 
greater  than  the  dimensions  of  the  molecules,  were 
still  much  larger,  and  that  the  chances  for  collisions 
of  more  than  two  molecules  at  a  time  were  so  small 
that  they  could  be  left  entirely  out  of  the  account. 
According  to  van  der  Waals  equation,  the  volume  in 
the  liquid  state  must  necessarily  be  less  than  the  criti- 
cal volume  3^,  b  itself  being,  as  we  have  found,  only 
4  times  the  volume  occupied  by  the  molecules  them- 
selves. From  this  we  see  that  if  the  space  were 
divided  up  evenly  among  all  the  molecules,  each  one 
could  have  a  cubical  space  whose  volume  could  not 
exceed  12  times  the  volume  of  the  molecule  itself,  that 

.174 


MOLECULES   WITHIN   A   LIQUID.  175 

is,  12  x  |7ro-3  =  27T0-3  and  whose  edge  consequently 
could  not  exceed  <r\^27r,  and  hence  must  be  consider- 
ably less  than  twice  the  diameter  of  the  molecule. 
It  is  evident  also  that  as  the  result  of  an  indefinite  in- 
crease of  pressure,  external  or  internal,  the  limiting 
volume  of  the  liquid  would  be  not  b  but  the  smallest 
space  into  which  the  molecules  could  be  packed,  which, 
if  the  molecules  were  spherical  would  be  a  little  more 
than  \b,  approximately  \b. 

There  is  evidence  also  that  while  in  most  gases  the 
composition  of  the  molecules  is  usually  represented 
fairly  accurately  by  the  ordinary  formulae,  in  many  if 
not  most  liquids  the  molecules  are  more  complex, 
being  made  up  of  two  or  more  -of  the  simple  gas- 
molecules  united,  and  hence  are  both  larger  and  less 
numerous,  so  that  even  if  the  same  equations  applied 
the  numerical  constants  must  be  different,  while  the 
latent  heat  of  vaporization,  so  called,  must  include  also 
some  latent  heat  of  dissociation.  This  cause  of  varia- 
into  will  be  taken  up  in  Chapter  XL 

While  we  have  then  a  fairly  satisfactory  discussion 
of  the  relations  of  pressure,  covolume,  temperature  and 
kinetic  energy  of  the  molecules  of  a  liquid,  and  of  the 
phenomena  of  vaporization,  we  still  have  to  discuss 
the  motions  of  the  molecule  within  the  liquid,  and  find 
if  possible  its  mean  free  path. 

Mean  Free  Path.  — We  shall  first  assume,  as  in  the 
case  of  gases,  that  the  molecules  are  all  spherical,  of 
diameter  cr,  and  that  all  but  the  one  we  are  especially 
considering  are  at  rest.  But  because  the  molecules 


76 


KINETIC   THEORY. 


are  so  near  together  we  shall  assume  a  particular  dis- 
tribution, such  that  the  centers  of  adjacent  molecules 
are  situated  at  the  vertices  of  equilateral  triangles. 
This  arrangement  gives  the  smallest  volume  when  all 
the  molecules  are  in  contact,  giving  a  total  volume  of 
about  J$,  as  we  have  just  stated,  or  ^  of  the  critical 


Fig.  16. 

volume,  hence  the  distance  between  the  centers  of 
adjacent  molecules  cannot  exceed  cr\/9,  and  hence 
will  usually  be  considerably  less  than  2cr.  Fig.  i6l 
will  represent  a  section  through  the  centers  of  seven 
such  molecules.  Consider  the  molecule  in  the  middle 
as  the  one  to  be  studied,  and  about  the  others  describe 
spheres  with  radius  a.  These  spheres  will  intersect, 

!Jager,  Wien.  Ber.,  102,  p.  257. 


MOLECULES   WITHIN   A    LIQUID. 


177 


since  the  distance  between  centers  is  less  than  2<r, 
leaving  a  small  volume  represented  by  the  shaded 
space  in  the  figure,  about  the  center  of  the  moving 
molecule.  This  small  volume  is  the  region  in  which 
the  center  of  the  molecule  is  free  to  move,  and  in 
which  we  must  find  its  mean  path.  We  can  for  our 
purposes  consider  this  space  spherical,  of  radius  h, 
(Fig.  17).  Our  problem  is  now  to  find  the  average 
length  of  the  path  from  any  point  of  the  surface  of 


Fig.  17. 

this  spherical  space  to  any  other  point  of  it,  that  is,  to 
find  the  average  value  of  the  line  hv  making  the  angle 
$  with  the  diameter  of  the  sphere.  We  have  found 
previously  (p.  34,  Eq.  (15)  )  that  the  proportion  of 
such  lines  making  an  angle  between  &  and  $  -f  dd- 
is  sin  &  dd-  while  the  length  of  such  a  line  is  evidently 
2h  cos  $,  hence  the  average  value  will  be 

flT/2 
2h  cos  &  sin  $d$  =  h  [sin2  #]  vl2  =  h. 
-  , 

12 


1  78  KINETIC   THEORY. 

This  value,  being  based  upon  the  assumption  that  all 
the  molecules  except  the  one  were  at  rest,  has  still  to 
be  multiplied  by  the  ratio  c/r,  which  we  call  as  before 
},  giving 


We  still  have  to  find  the  value  of  7z,  which  evidently 
cannot  vary  much  from  the  radius  of  the  sphere  de- 
scribed in  the  free  space  about  the  center  of  the  mov- 
ing molecule.  This  last  easily  appears  to  be  d  —  cr 
where  d  represents  the  mean  distance  between  the 
centers  of  adjacent  molecules. 

We  may  find  the  value  of  d  by  the  following  device  : 
Suppose  the  space  v,  which  contains  N  molecules,  to  be 
rectangular  in  shape  ;  along  one.  edge  the  molecules  are 
placed  regularly  at  intervals  d  from  center  to  center  ; 
suppose  the  number  in  the  row  to  be 
nr  Then  a  second  row  of  n^  mole- 
cules is  placed  so  that  the  centers 
of  three  adjacent  molecules  will  be 
at  the  vertices  of  an  equilateral  tri- 
angle of  edge  d.  The  distance  be- 
tween the  lines  of  centers  will  be 
the  altitude  of  this  triangle,  which  is  \d\/  '3.  Sup- 
pose there  can  be  placed  in  all  nt  such  rows.  Then 
above  this  layer  place  another  layer,  so  that  each 
molecule  will  have  its  center  at  the  same  distance  d 
from  the  centers  of  the  three  nearest  molecules  of 
the  lower  layer,  the  four  centers  being  thus  at  the 
vertices  of  a  regular  tetrahedron  whose  edge  is  d, 


MOLECULES   WITHIN   A   LIQUID. 

and  whose  altitude  must  be  %d{/6.  Suppose  the 
total  number  of  such  layers  in  the  space  to  be  ny 
Then  the  total  volume  will  be 


but  this  volume  is  v  ;  and  njiji^  is  the  total  number 
of  molecules,  Ny  hence 


v  = 


The  minimum  volume,  which  we  may  call  bv  is 
found  by  letting  d  =  a, 


from  which 


Substituting,  we  have 


The  space  actually  occupied  by  the  molecules  is  (p. 
69) 


This  was  according  to  the  deduction  of  van  der  Waals' 
equation  \b,  and  hence  ^  the  critical  volume,  from 
which  we  find 


l8o  KINETIC   THEORY. 

The  greatest  possible  value  of  d,  and  hence  of  /,  would 
be  for  the  critical  state,  in  which  vc  =  3^,  and  hence 


=  2.O70-, 
113 

/=  £(2.07er  -0-)  =  .8(7. 

For  v  ^=  2b  which  is  for  van  der  Waals'  equation  a  fair 
average  value  of  the  volume,  and  which  for  Dieterici's 
equation,  which  we  have  found  to  agree  remarkably 
well  with  some  of  the  properties  of  the  critical  state, 
is  the  critical  volume,  we  have 


==  I6 

v         2b 
d  =  1.640-, 
/  =  .480-. 

From  these  we  conclude  that  the  critical  volume  is 
not  more  than  9  (or  6)  times  the  minimum  attainable 
volume,  that  the  mean  distance  between  the  centers 
of  adjacent  molecules  can  only  in  the  extreme  case 
exceed  twice  their  diameters,  but  is  ordinarily  much 
less  ;  that  the  mean  free  path  is  less,  usually  less  than 
half  this  diameter,  having  of  course  for  its  minimum 
value  o  when  the  molecules  are  in  contact,  and  d=o-. 

Pressure,  —  The  average  number  of  impacts  per 
second  of  such  a  moving  molecule  is  evidently  cjl. 
Suppose  one  particular  molecule  moving  in  its  free 
space  with  the  speed  c,  making  cjl  impacts  per  second 


MOLECULES  WITHIN  A  LIQUID.  181 

and  having  a  momentum  of  me ;  then  the  total  force 
required  to  hold  it  in  this  space  will  be  the  product 
me*  1 1.  The  area  on  which  this  force  is  exerted  is  the 
area  of  the  little  spherical  space  of  radius  h  and  area 
4rf,  so  that  the  pressure  will  be  me* / ^rf/.  Passing 
to  averages,  and  expressing  the  total  pressure  by  p  -f  P 
we  have 


or  substituting  for  /  its  value  |^, 


Multiplying  by  irNh^  we  have 
(p  +  P)  TrNtf 
which  is  of  the  form 

RT 


given  in  Chapter  VII.  It  readily  appears  that  the  sum 
of  all  the  free  spaces  of  volume  |?r^3  about  each  of 
the  N  molecules  is  |7rA%3,  and  hence  that  the  covolume 
irN/fi  is  |  of  the  sum  of  these  free  spaces,  the  factor 
J  being  the  ratio  c/r,  used  in  finding  the  mean  free 
path.  It  is  evident  from  the  previous  discussion  that 
h  cannot  except  in  the  extreme  case  of  the  critical 
volume  exceed  \d,  and  hence  that  this  covolume  TrNh* 
cannot  exceed  fytNd*=  1/2/877-?;  =  .572;,  that  is,  the 
covolume  is  ordinarily  less  than  half  the  volume  of  the 


1 82  KINETIC   THEORY. 

liquid,  usually  much  less,  e.  g.,  if  v  =  2&,  |  the  critical 
volume  or  for  Dieterici's  equation  the  critical  volume, 
the  covolume  is  .27^,  about  J.  This  gives  the  covol- 
ume  a  physical  meaning,  although  its  analytical  form 
is  different  from  that  given  by  van  der  Waals'  equa- 
tion, v  —  b,  in  which  b  is  a  constant.  Eliminating  h  by 
the  equations 

h  =  d  —  <r  =  d(  i  —  -, 


we  get 

f  f  /T    \ 

or 


which  is  expressed  explicitly  in  terms  of  the  volume 
of  the  liquid  and  the  space  occupied  by  the  molecules. 

For  Dieterici's  equation  the  critical  volume  is  2b, 
and  hence,  as  we  have  found  above,  d  cannot  exceed 
1.6420-,  /  cannot  be  greater  than  .4820-,  and  <E>  can- 
not be  greater  than  .27?',  so  that  a  substance  follow- 
ing Dieterici's  equation  in  the  gaseous  state  ought  to 
have  its  behavior  in  the  liquid  state  in  close  harmony 
with  the  results  of  the  present  discussion. 

Internal  Pressure.  —  Approximate  values  of  the  in- 
ternal pressure  of  liquids  have  been  given  in  Chap- 
ter VII.  Some  of  these  were  based  upon  assumptions 
as  to  the  covolume  of  the  liquid  which  we  did  not 


MOLECULES   WITHIN   A   LIQUID.  183 

there  attempt  to  verify  ;  others  were  obtained  by  the 
use  of  the  formula  ajv*,  which  can  hardly  apply  in 
this  state,  although  its  form  was  derived  from  a  study 
of  the  theory  of  surface  tension  and  capillarity.  Their 
approximate  agreement  would  nevertheless  seem  to 
indicate  that  we  have  at  least  learned  the  order  of 
magnitude  of  this  pressure.  The  formula  deduced  for 
the  latent  heat  might  also  be  used,  if  only  we  could 
readily  find  the  covolume  in  the  liquid  state. 

Since  this  internal  pressure  is  but  the  equilibrant  of 
the  forces  at  the  surface,  which  determine  the  volume 
of  the  liquid  and  manifest  themselves  in  the  phenomena 
of  surface  tension,  it  must  have  the  same  temperature 
coefficient,  and  hence  can  be  expressed  by  the  formula 
P=  PQ(i  —  et)  in  which  PQ  represents  the  value  of  P 
at  o°  C.,  and  e  is  the  temperature  coefficient  of  surface 
tension.  Since  e  is  always  positive  it  appears  that  with 
rising  temperature  the  internal  pressure,  but  not  the 
total  pressure,  decreases.  Our  general  equation  of 
condition  may  similarly  be  written,  neglecting  the 
external  pressure, 


Dividing  this  by  the  last  we  get 


which  shows,  since  a  and  e  are  both  positive,  that 
must  increase  rapidly  with  the  temperature. 


1  84  KINETIC   THEORY. 

The  coefficient  of  compressibility  is  defined  as  the 
ratio  of  the  relative  decrease  of  volume  to  the  pressure 
producing  the  decrease.  Analytically  it  is 

I   dv 

~ 


differentiating  the  equation 


on  the  assumption  that  T  is  constant,  we  obtain,  after 
simplification 


idv 

-= —  > 


vdp          p+P        (/- 
or  dropping  /  and  solving  for  Py 

/>-!-*- 

K  1/2  ITU 

which  still  requires  a  knowledge  of  <I>  or  of  the  ratio 

V  2TTV         \  d) 

to  find  P.  The  minimum  values  for  ffjd,  and  the 
maximum  values  for  K  which  would  give  a  lower 
limit  for  Py  would  of*  course  be  found  in  the  critical 
state,  for  which  (i  —  °"A^)3  can  easily  be  determined. 
Approximate  numerical  results  might  perhaps  be 
obtained  by  computing  the  covolume  in  terms  of  v 
and  blt  determining  the  latter  by  finding  the  minimum 
volume  to  which  the  liquid  could  be  compressed  by 
the  most  intense  pressure  which  could  be  applied. 


CHAPTER    IX. 
SOLUTIONS. 

MIXTURES  and  solutions  may,  like  pure  substances, 
exist  in  either  the  gaseous,  the  liquid,  or  the  solid 
state.  The  elementary  laws  of  mixtures  of  gases 
which  do  not  react  chemically  with  each  other  have 
already  been  developed  (Chap.  II.,  pp.  38-43).  When 
equilibrium  is  attained  the  mean  kinetic  energy  of 
translation  of  the  molecules  is  the  same  for  each  of 
the  gases  composing  the  mixture,  and  most  of  the 
other  properties  are  additive  ;  for  instance  the  pressure 
exerted  is  the  sum  of  all  the  pressures  that  would  be 
exerted  by  the  different  gases  if  each  were  present 
separately,  occupying  the  whole  volume,  and  the  en- 
ergy is  the  sum  of  the  energies  of  all  the  component 
gases.  In  general  we  may  say  that  gases  and  vapors 
mix  in  all  proportions,  and  the  equations  representing 
the  behavior  of  the  mixtures  are  of  the  same  type  as 
those  referring  to  pure  substances.  The  question 
arises  with  regard  to  the  entropy  of  a  mixture  of 
gases,  whether  it  is  to  be  regarded  as  the  sum  of  the 
entropies  of  the  different  components  each  regarded 
as  occupying  all  the  volume  with  its  appropriate  par- 
tial pressure  ;  or  regarded  as  occupying  their  propor- 
tional parts  of  the  volume,  with  a  pressure  equal  to 
that  of  the  mixture.  This  latter  view  is  evidently  the 

185 


1 86 


KINETIC   THEORY. 


correct  one  in  the  case  of  successive  additions  of  por- 
tions of  a  single  gas,  as  after  the  volumes  are  brought 
into  contact  with  each  other  diffusion  takes  place 
spontaneously  without  the  expenditure  of  energy  or 
the  absorption  of  heat ;  while  allowing  each  portion 
to  fill  the  whole  available  space,  the  addition  of  each 
successive  increment  will  demand  the  expenditure  of 
energy,  in  compressing  both  the  portion  of  gas  al- 
ready present,  and  that  being  added,  and  hence  there 
will  result  a  change  either  of  the  temperature  or  of 


Fig.  19. 

the  entropy.  Suppose,  however,  two  different  gases 
at  the  same  pressure  and  temperature  are  placed  in 
contact  and  allowed  to  diffuse  into  each  other ;  this 
process  can  take  place  without  expenditure  of  energy 
and  without  absorption  of  heat,  but  we  can  also  im- 
agine the  diffusion  to  take  place  by  a  method  which 
will  involve  the  expenditure  of  work,  and  hence,  if 
the  temperature  be  kept  constant,  with  an  absorption 
of  heat  and  a  change  in  the  entropy.  Suppose  that 
in  a  cylinder  we  have  two  gases,  which  we  will  call  A 
and  B,  and  two  movable  pistons,  one  of  which,  say 
the  one  to  the  left,  will  allow  the  gas  A  to  pass 


SOLUTIONS.  187 

through  it  freely,  while  it  is  perfectly  impervious  to 
the  gas  B.  Similarly  the  one  to  the  right  we  may 
consider  permeable  to  the  gas  B,  but  impervious  to  A. 
We  shall  consider  the  two  pistons  to  be  originally  in 
contact  with  each  other,  with  the  gas  A  to  the  left 
and  B  to  the  right,  the  positions  of  the  pistons  being 
so  chosen  that  the  two  gases  are  at  the  same  pres- 
sure. Now  the  piston  at  the  left,  being  perfectly  per- 
meable to  the  gas  A  experiences  when  at  rest  no 
excess  of  force  from  A  tending  to  move  it  in  either 
direction,  while  the  gas  B,  passing  through  the  other 
piston,  exerts  its  full  pressure  upon  it,  hence  there  is 
a  tendency  for  this  piston  to  be  pushed  by  the  gas  B 
to  the  extreme  left  hand  end  of  the  cylinder,  doing 
upon  it  an  amount  of  work  just  the  same  as  if  the  gas 
A  were  not  present  and  hence,  if  the  temperature  be 
kept  constant,  absorbing  an  amount  of  heat  just 
equivalent  to  this  work.  Similarly  the  gas  A  will 
tend  to  push  the  piston  at  the  right  to  the  extreme 
right  hand  end  of  the  cylinder,  doing  upon  it,  if  the 
temperature  be  kept  constant,  the  amount  of  work  it 
would  do  in  the  same  isothermal  expansion  of  the  gas 
B  were  not  present,  and  absorbing  the  corresponding 
amount  of  heat.  When  both  these  processes  have 
been  accomplished  the  diffusion  is  complete,  but  each 
amount  of  gas  has,  during  the  diffusion,  absorbed  at 
constant  temperature  a  certain  amount  of  heat,  and 
hence  changed  its  entropy  by  a  certain  amount.  It  is 
easy  to  see  that  for  each  gas  this  change  is  just  the 
same  as  that  which  would  take  place  if  it  were  al- 


1 88  KINETIC   THEORY. 

lowed  to  expand  from  its  original  volume  and  the 
pressure  of  the  mixture  of  gases  to  the  full  volume 
of  the  mixture  and  the  partial  pressure  which  it  there 
exerts.  The  entropy  of  a  substance  depends  only 
upon  its  state,  hence  we  conclude  that  the  entropy  of 
a  mixture  of  gases  is  the  sum  of  the  entropies  which 
each  would  have  if  occupying  the  whole  volume  of 
the  mixture  at  its  appropriate  partial  pressure.  This 
method  of  treatment  evidently  cannot  be  applied  to 
different  portions  of  the  same  gas  so  that  our  previous 
conclusion  will  still  hold,  that  the  entropy  of  a  large 
body  of  a  single  gas  is  the  sum  of  the  entropies  of 
the  smaller  volumes  of  which  it  is  made  up,  each  at 
the  pressure  of  the  whole.  The  process  which  we 
have  just  considered  is  an  ideal  one.  The  transfor- 
mation we  have  described  is  evidently  reversible, 
hence  no  objections  to  it  can  be  raised  on  that  score. 
Its  validity  then  depends  upon  the  possibility  of  being 
able  to  realize  such  pistons  which  shall  be  permeable 
to  one  gas,  but  not  to  another.  We  know  that  hot 
palladium  allows  hydrogen  to  pass  through  it  with 
considerable  freedom,  while  ammonia  gas,  on  account 
of  its  great  solubility  in  water,  will  pass  with  consider- 
able freedom  through  a  wet  membrane,  hence  such  a 
system  might  be  realized  for  these  two  gases.  Since 
we  believe  in  the  uniformity  of  the  laws  of  nature  we 
then  accept  this  theorem  as  general. 

The  study  of  solid  solutions,  such  as  alloys,  mixed 
crystals,  etc.,  and  of  the  phenomena  of  diffusion  in 
solids,  has  attracted  much  attention  in  recent  years, 


SOLUTIONS.  189 

and  has  yielded  some  results  of  value,  but  is  beyond 
the  scope  of  this  work. 

Liquid  solutions  may  be  classified  according  to  the 
state  of  the  components  either  before  the  solution,  or 
when  they  have  been  separated.  Thus  we  speak  of 
solutions  of  a  gas,  of  a  liquid,  or  of  a  solid.  The  sim- 
plest case  of  a  liquid  solution  is  one  in  which  one 
component  is  so  much  more  volatile  than  the  other 
that  the  vapor  above  the  solution  may  be  regarded  as 
a  pure  vapor.  This  may  be  done  when  the  dissolved 
substance  is  a  gas,  in  which  case  the  vapor  of  the  sol- 
vent is  disregarded,  or  when  the  dissolved  substance 
is  a  non-volatile  liquid  or  solid,  in  which  case  the 
vapor  of  the  solvent  only  is  considered. 

Absorption  of  Gases,  —  The  amount  of  a  gas  which 
a  given  liquid  can  dissolve  has  been  found  by  experi- 
ment to  be  proportional  to  the  pressure  of  the  gas 
upon  the  surface  of  the  liquid ;  or  in  other  words,  the 
quantity  of  the  gas  contained  in  a  given  volume  of  the 
liquid  bears  a  definite  ratio  to  the  quantity  contained 
in  the  same  volume  of  the  free  space  above  it  which 
is  independent  of  the  pressure  and  depends  at  any 
given  temperature  only  upon  the  liquid  and  gas  con- 
cerned. This  statement,  which  is  commonly  known 
as  HENRY'S  LAW,  expresses  with  reasonable  accuracy 
the  behavior  of  many  gases ;  it  is  not  strictly  true  in 
all  cases,  however,  and  hence  is  to  be  regarded,  like 
Boyle's  Law,  as  a  convenient  and  useful  first  approxi- 
mation. The  difference  in  the  solubilities  of  different 
gases  is  shown  by  the  following  values  of  the  coef- 


190  KINETIC   THEORY. 

ficient  of  absorption,  that  is  the  ratio  of  the  volume  of 
the  gas  absorbed  to  the  volume  of  the  absorbing  sub- 
stance, selected  from  data  given  by  Bunsen.1 

Substance.  Solvent. 

Water.  Alcohol. 

Ammonia,  NH3  727.2 

Sulphur  dioxide,  SO2  43. 56  144-55 

Sulphuretted  hydrogen,  H2S  3-233  9-54 

Nitrous  oxide,  N2O  .778  3.268 

Carbon  dioxide,  CO2  1.002  3.2 

Carbon  monoxide,  CO  .0243  .2044 

Oxygen,  O2  .03  .284 

Nitrogen,  N2  .0145  .1214 

Air,  .01 79 

Hydrogen,  H2  .0193  .0673 

It  is  readily  seen  that  gases  which  are  more  strongly 
absorbed  by  one  liquid  are  in  general  also  more 
strongly  absorbed  by  the  other  liquid,  but  there  seems 
to  be  no  simple  and  universal  law  relating  the  different 
solubilities. 

As  has  been  stated,  Henry's  law  is  to  be  regarded 
as  simply  a  first  approximation.  That  the  coefficient 
of  absorption  should  be  independent  of  the  pressure 
could  be  reasonably  expected  in  the  case  of  the  less 
soluble  gases  for  moderate  pressures.  That  this  is  true 
is  shown  by  the  following  tables2  for  the  absorption 
of  carbon  dioxide  in  water,  in  which  the  column 
marked  p  gives  the  pressure  in  mm.  of  mercury  and  a 
the  coefficient  of  absorption : 

1  For  a  more  complete  discussion,  with  reference  to  the  literature  on 
the  subject  and  tabulations  of  the  results  see  Winkelmann,  "  Handbuch 
der  Physik,"  I.,  pp.  669-682,  Article  "Absorption." 

SKhanikoff  &  Longuinine,  Ann.  Ch.  Ph.  (4),  II,  p.  412,  1866. 


SOLUTIONS.  191 


P 

a 

P 

a 

P 

a 

697.7 

1.0289 

2002.1 

1.1037 

2738.3 

I.IIIO 

809.0 

1.0908 

2188.7 

I.I023 

3109.5 

I.IOOO 

1289.4 

I.I247 

2369.0 

I.II82 

1469.9 

I.II79 

2554.0 

I.I055 

While  the  variations  in  the  value  of  a  are  consider- 
able, from  their  irregularities  they  seem  to  be  due  to 
experimental  error  only.  In  the  case  of  the  absorp- 
tion of  sulphur  dioxide  by  water,  the  variation  is 
somewhat  more  marked  and  more  regular  as  is 
shown  by  the  following  table,1  in  which  /  gives  the 
pressure  in  meters  of  mercury,  g  the  weight  of  gas 
absorbed  and  gjp  the  ratio  of  the  two : 

/         .05         .1  .2         .5          i.  1.3 

g         .015        .027        .05        .118        .229        .295 
g\p        .3  .27          -25        -24          -23          .23 

While  in  the  case  of  the  absorption  of  ammonia  by 
water  the  table2  shows  a  very  marked  change. 

p  .06  .1  .2  .5  I.  2. 

g  .119  .158  .232  .403  .613  .982 

g\6  2.  1.6  1.2  .8  .6  .5 

It  is  naturally  to  be  expected  that  the  coefficient  of 
absorption  should  be  dependent  upon  the  temperature  ; 
in  fact  it  is  a  matter  of  common  experience  that  water 
can  be  freed  of  air  by  heating,  and  to  a  greater  extent 
than  would  be  accounted  for  by  the  simple  expansion 
of  air.  The  decrease  in  the  solubility  is  not  propor- 

1  Sims,  Ann.  Pharni.,  118,  p.  334,  1861. 

2Roscoe  &  Dittmar,  Ann.  Pharm.,  112,  p.  349,  1859. 


I92 


KINETIC   THEORY. 


tional  to  the  change  of  temperature  but  can  be  ex- 
pressed with  fair  accuracy  by  the  formula 

a  =  A  -  Bt  +  Ct\ 

The  following  table  gives  the  value  of  A,  B  and  C 
according  to  Bunsen  for  water  and  alcohol,  for  some 
of  the  gases  of  the  previous  table,  between  o°  and 
20°  C. 


WATER. 

ALCOHOL. 

A. 

B. 

C. 

A. 

B. 

C. 

S02  79-8 

2.6l 

.0293 

328. 

16.8 

.8 

H2S  4.37 

.0837 

.000521 

17.89 

.656 

.0066 

N2O  1.305 

•0454 

.000684 

4.178 

.0698 

.00061 

C02  1.7967 

.0776 

.00164 

4-33 

.094 

.00124 

CO   .0329 

.000816 

.0000164 

.204 

O2    .04115 

.00109 

.0000226 

.284 

N2   .0203 

.000539 

.00001116 

.1263 

.00042 

.000006 

H2   .0193 

.0691 

.00015 

.000001 

Solution  of  Liquids.  —  We  are  familiar  with  the  fact 
that  some  liquids  will  mix  in  all  proportions,  some  not 
at  all,  and  others  in  all  proportions  up  to  a  certain 
limit.  We  may  take  as  examples  water  and  alcohol, 
which  mix  in  all  proportions ;  water  and  oil,  which 
do  not  mix ;  and  water  and  ether,  which  when  shaken 
up  together  and  then  allowed  to  come  to  rest  separate 
into  two  layers,  the  upper  of  ether  saturated  with 
water,  the  lower  of  water  saturated  with  ether.  We 
may  represent  the  phenomena  in  the  case  of  substances 
with  limited  solubility  graphically  by  a  diagram  such 
as  that  of  Fig.  20  where  the  abscissa  represents  the 
temperature  of  the  solution,  and  the  ordinate  repre- 
sents the  proportional  parts  of  one  of  the  components, 


SOLUTIONS.  193 

in  per  cents  for  convenience,  while  the  proportional 
part  of  the  other  component  is  given  by  the  distance 
measured  vertically  downward  from  the  horizontal 
line,  the  sum  of  the  two  being  the  whole  amount  of 
the  solution,  100  per  cent.  For  such  a  mixture  as 
water  and  ether,  which  have  two  proportions  of  solu- 
bility, there  will  be  for  a  given  temperature  two  points, 


A     B 

0     100 


50     50 


!00     0 


Fig.  20. 


or  in  general,  the  mutual  solubilities  of  the  two  liquids 
at  various  temperatures  will  be  expressed  by  two 
curves,  which  as  the  temperature  increases  gradually 
approach  each  other,  and  for  many  substances  have 
been  found  experimentally  to  meet,  as  shown  in  the 
figure ;  at  or  above  the  temperature  represented  by 
this  point  the  two  components  will  dissolve  each  other 
in  all  proportions. 

The  vapor  over  the  surface  of  a  mixture  or  solution 
will  in  general  be  a  mixture  of  the  Vapors  of  the  two 
components  of  the  solutions,  but  not  necessarily  or 
generally  in  the  same  proportions  as  those  which  pre- 
vail in  the  solution.  Very  little  work  has  been  done 
so  far  in  the  way  of  an  experimental  determination  of 
the  composition  of  such  vapors  over  mixtures.  Much 


194  KINETIC   THEORY. 

study  has  been  given,  however,  to  the  determination 
of  the  tension  of  the  vapor  over  mixtures.  The  ordi- 
nary effect  of  the  addition  of  a  small  quantity  of  a  sec- 
ond volatile  liquid  to  a  first  is  to  change  the  tension 
of  the  vapor  above  it  not  by  a  large,  but  by  a  small 
amount.  The  presence  of  the  vapor  of  the  second 
liquid  in  addition  to  that  of  the  first  will  tend  to  in- 
crease the  tension  of  the  resulting  vapor,  while  if  the 
second  liquid  has  quite  a  marked  affinity  for  the  vapor 
of  the  first,  its  vapor  will  be  present  in  less  amount ; 
at  any  rate,  the  addition  of  the  small  amount  of  the 
second  liquid  in  some  cases  increases  the  tension  of 
the  vapor,  in  some  decreases  it,  but  always  by  a  small 
amount. 

The  tension  of  the  vapor  over  a  mixture  of  liquids 
can  never  exceed  the  sum  of  the  tensions  of  the  vapor 
of  each  liquid  taken  separately  at  that  temperature. 
This  follows  immediately  from  the  conceptions  of  the 
kinetic  theory,  by  reference  to  the  theory  of  vaporiza" 
tion.  In  the  process  of  vaporization,  as  in  the  other 
states  of  matter,  the  only  forces  acting  on  the  mole- 
cules are  those  which  we  associate  with  the  collisions, 
and  the  mutual  attractive  forces.  The  attractions  be- 
tween molecules  appear,  from  all  our  experience  of 
them,  to  be  selective,  so  that  molecules  of  the  same 
kind  appear  to  exert  mutual  attractive  forces,  while 
molecules  of  different  kinds  may  or  may  not  seem  to 
attract  each  other ;  repulsions  do  not  seem  to  occur 
except  during  the  collisions.  In  a  mixture  of  liquids 
the  two  kinds  of  molecules  will  in  general  have  some 


SOLUTIONS.  195 

mutual  attraction,  otherwise  the  two  liquids  would 
tend  to  separate.  This  may  be  one  reason  why 
liquids  ordinarily  shrink  in  mixing.  A  molecule  in 
trying  to  pass  through  the  non-homogeneous  layer 
will  then  in  general  find  itself  nearer  on  the  average 
to  other  attracting  molecules,  and  subject  to  as  great 
or  greater  forces  tending  to  return  it  to  the  body  of 
the  liquid  than  if  the  molecules  of  its  own  kind  only 
were  present.  Hence  the  energy  which  such  mole- 
cules must  have  in  order  to  escape  is  greater  than  in 
the  case  of  the  pure  liquid,  and  the  number  of  mole- 
cules above  the  liquid  and  the  partial  pressure  due  to 
them  less  than  in  the  case  of  the  vapor  over  the  pure 
liquid. 

This  may  also  be  shown  as  a  consequence  of  thermo- 
dynamical  considerations.  If  the  tension  of  the  vapor 
over  the  mixture  be  greater  than  the  sum  of  the  ten- 
sions of  the  pure  vapors  over  their  pure  liquid  at  the 
same  temperature,  then  at  least  one  of  the  vapors  must 
have  its  partial  tension  greater  than  its  tension  over 
its  pure  liquid.  If  then  this  vapor  be  in  contact  with 
the  mixture  and  with  a  body  of  the  pure  liquid,  at  the 
surface  of  the  latter  it  will  begin  to  condense,  tending 
to  reduce  its  tension  to  that  giving  equilibrium  at  that 
surface.  This  lowering  of  the  tension  will  result  in 
further  evaporation  of  this  component  from  the  mix- 
ture, thus  giving  an  automatic  process  of  separation 
of  this  component,  and  since  the  process  of  diffusion 
of  a  pure  substance  into  a  mixture  is  one  which  takes 
place  of  itself,  we  have  a  cycle  of  processes  which 


196 


KINETIC   THEORY. 


could  take  place  spontaneously  and  hence  be  made  to 
do  work,  at  a  fixed  temperature.  That  such  an  iso- 
thermal cycle  should  do  useful  work  is,  since  the  maxi- 
mum amount  of  work  which  can  be  done  in  any  cycle 
is  proportional  to  the  difference  of  the  extreme  tem- 
peratures of  the  cycle,  in  this  case  zero,  contrary  to 
our  general  experience  as  expressed  in  the  Second 


Fig.  21. 

Law  of  Thermodynamics.  Hence  the  assumption 
that  the  vapor  tension  over  a  mixture  of  two  liquids 
is  greater  than  the  sum  of  the  separate  tensions  of  the 
component  vapors  over  their  pure  liquids  leads  to  in- 
admissible consequences,  and  is  itself  inadmissible. 

The  relation  of  the  vapor  tension  to  the  composition 
of  the  liquid  is  readily  shown  and  studied  by  means 
of  diagrams  in  which  the  abscissas  represent  the  pro- 
portion of  one  of  the  components,  that  of  the  other  com- 


SOLUTIONS.  197 


ponent  being  similarly  measured  from  the  other  end  of 
the  diagram,  while  the  ordinates  represent  the  tension. 
We  then  obtain  curves  of  different  forms,  according 
as  the  addition  of  a  small  amount  of  either  liquid  to 
the  other  pure  component  tends  to  decrease  the  vapor 
tension,  (I) ;  or  to  increase  it,  (II) ;  or  the  addition 
of  the  second  to  the  first  tends  to  decrease  the  vapor 
tension,  while  the  addition  of  the  first  to  the  second 
increases  it,  (Ilia  and  Illb).  Curves l  like  case  I  are 
actually  observed  in  the  case  of  mixtures  of  formic 
acid  and  water ;  case  II  is  that  of  water  and  propyl 
alcohol ;  and  case  Ilia  that  of  water  and  either  ethyl 
or  methyl  alcohol.  Case  II Ib  has  so  far  never  been 
obtained  experimentally,  and  is  only  mentioned  for 
the  sake  of  completeness  of  statement. 

In  the  case  of  such  a  mixture  as  that  of  water  and 
ether,  which  separates  into  two  layers,  the  vapor  ten- 
sion is  the  same  over  either  layer,  the  water  saturated 
with  ether,  or  the  ether  saturated  with  water ;  for  it  is 
easy  to  place  them  in  a  U-shaped  tube  so  that  in  one 
branch  one  of  the  solutions,  and  in  the  other  the  other 
should  be  exposed  to  their  vapor,  while  within  the 
tube  the  two  should  be  in  contact.  Then  according 
to  the  considerations  adduced  a  little  while  ago,  there 
must  be  a  state  of  equilibrium,  with  the  vapor  tension 
the  same  over  both  surfaces,  or  we  should  have  an 
automatic  isothermal  cycle  from  which  we  could  obtain 
work,  a  thing  which  we  believe  impossible.  The  curve 
of  vapor  tension  over  such  a  solution  will  then  consist 

1  Konowalow,   Wied.  Ann.,  14,  p.  34,  1881. 


198  KINETIC   THEORY. 

of  a  straight,  horizontal  line  for  all  proportions  of  the 
mixture  between  the  two  stable  proportions  of  satura- 
tion, since  for  any  such  intermediate  proportions  the 
mixture  separates  into  the  two  parts,  and  the  tension 
does  not  depend  on  the  relative  amounts  of  these  which 
are  present. 

A  study  of  the  various  shapes  of  the  curves  of  vapor 
tension  helps  to  explain  the  phenomena  of  distillation, 
and  in  particular  of  fractional  distillation.  The  vapor 
over  a  mixture  of  two  liquids  will  in  general  be  richer 
than  the  liquid  phase  in  the  more  volatile  component ; 
if  this  vapor  be  condensed  and  then  redistilled,  its 
resulting  vapor  will  be  one  of  still  higher  tension,  or 
what  amounts  to  the  same  thing,  if  the  distillation  is 
at  constant  pressure  rather  than  at  constant  tempera- 
ture, which  is  the  more  common  case,  of  lower  boiling 
point  than  the  original  liquid.  By  repeated  distilla- 
tions the  liquid  is  thus  separated  into  two  components, 
one  more  volatile,  the  other  less  volatile.  If  the  curve 
of  vapor  tensions  be  of  the  form  Ilia  these  two  com- 
ponents will  be  the  two  pure  substances  ;  if,  however, 
the  curve  be  like  that  marked  I,  the  residual  compo- 
nent will  have  the  composition  corresponding  to  the 
minimum  vapor  tension,  or  least  volatility,  while  the 
other  component  will  be  the  pure  substance  which  is 
present  in  greater  amount  in  the  original  mixture  than 
in  this  residuum  ;  similarly  if  the  curve  be  of  the  form 
II,  the  final  distillate  will  be  the  mixture  having  highest 
vapor  tension,  and  the  residuum  will  be  the  pure  sub- 
stance present  in  the  original  mixture  in  excess. 


SOLUTIONS.  199 

Osmosis.  —  It  has  long  been  known  that  certain 
membranes  allow  the  passage  of  some  substances 
through  them,  while  they  are  completely  impervious 
to  others ;  healthy  lung  tissue  allows  the  oxygen  of 
the  air  free  access  to  the  blood  in  its  capillaries,  while 
completely  retaining  the  blood  itself;  plants  and  flowers 
are  freshened  by  placing  them  in  water,  without  any 
appreciable  loss  of  their  own  soluble  constituents. 
Wishing  to  obtain  further  insight  into  the  phenomena 
of  osmosis  in  plant-cells,  that  is,  the  passage  of  water 
into  and  out  of  them,  the  botanist,  W.  Pfeffer,1  at- 
tempted to  imitate  on  a  large  scale  the  construction 
of  a  plant  cell.  If  we  separate  a  body  of  a  solution 
and  its  pure  solvent  by  a  portion  through  which  the 
solvent  can  pass,  but  not  the  dissolved  substance,  the 
solvent  will  diffuse  through  the  partition,  thus  pro- 
ducing an  excess  of  material,  and  hence  a  considerable 
pressure  on  the  partition.  The  excess  of  pressure  of 
the  solution  over  that  of  the  solvent  is  called  Osmotic 
Pressure.  In  making  a  satisfactory  cell  for  the  study 
of  osmotic  phenomena  two  things  are  necessary :  that 
it  should  be  completely  impervious  to  the  dissolved 
substance  while  allowing  the  solvent  to  pass  through 
it,  and  that  it  should  have  the  mechanical  strength  to 
sustain  whatever  pressure  it  may  need  to  be  subjected 
to.  These  conditions  seem  to  be  best  satisfied  by  cer- 
tain films  of  precipitation,  and  particularly  by  a  film 
of  copper  ferrocyanide.  The  phenomena  of  osmosis 

i  "  Osmotische  Untersuchungen, "  Leipzig,   1877,  Harper's  Science 
Series,  IV.,  p.  3. 


200  KINETIC   THEORY. 

can  be  illustrated  qualitatively  by  placing  in  a  cup  of 
unglazed  earthenware,  such  as  is  used  to  separate  the 
liquids  in  some  types  of  battery  cells,  a  solution  of 
sugar  containing  a  little  copper  sulphate,  closing  this 
cup  with  a  well-fitting  cork,  carrying  a  long  slim  glass 
tube  and  well  sealed  with  sealing  wax,  and  immersing 
the  cup  completely  in  a  weak  solution  of  potassium 
ferrocyanide.  The  copper  sulphate  and  potassium  fer- 
rocyanide,  meeting  in  the  walls  of  the  cup,  will  form 
a  film  of  insoluble  copper  ferrocyanide  which  will  pre- 
vent the  passage  of  the  dissolved  substances,  but  not 
of  the  water.  If  then  the  apparatus  be  allowed  to 
stand  for  some  time  water  will  gradually  enter  the 
cup,  causing  the  solution  to  rise  in  the  tube,  and  thus 
showing  that  there  is  an  excess  of  pressure  in  the 
inside  of  the  cup.  For  quantitative  work  it  is  neces- 
sary to  prepare  the  semi-permeable  membrane,  as  it 
is  called,  with  much  greater  care,  since  the  pressures 
observed  may  be  of  considerable  magnitude  and  to 
replace  the  open  tube  of  liquid  by  a  form  of  manom- 
eter which  shall  be  able  to  register  a  much  greater 
range  of  pressure,  while  allowing  only  a  slight  increase 
in  the  volume  of  the  liquid,  and  hence  only  a  negli- 
gible dilution  of  the  solution.  For  a  more  detailed 
description  of  methods  and  immediate  results  the  reader 
is  referred  to  the  original  paper. 

Osmotic  Pressure.  — The  phenomena  of  osmosis  and 
osmotic  pressure  can  be  observed  to  best  advantage 
because  least  complicated  by  other  phenomena  in 
dilute  solutions  of  non-volatile  substances.  Cane 


SOLUTIONS.  201 

sugar  has  been  found  a  convenient  material  and  a  few 
of  Pfeffer's  results  for  it  are  here  given. 

Osmotic  Pressure  for  Cane  Sugar  of  Different  Con- 
centration. 

Percentage  Cone. 


By  Weight. 

Osmotic  Pressure. 

Pressure/Cone. 

1.0 

535  mm. 

535 

2.0 

1016  mm. 

506 

2.74 

1518  mm. 

554 

4.0 

2082  mm. 

521 

6.0 

3075  mm. 

5i3 

Effect  of  Temperature  on  Osmotic  Pressure.  —  The 
following  results  were  obtained  with  a  I  per  cent,  solu- 
tion of  cane  sugar  : 

Temperature.  Pressure.  Comp.  Pressure. 

14.2°  C.  5 10  mm.  519 

32.0°  C.  544mm.  551 

6.8°  C.  505  mm.  505 

13.7°  C.  525  mm.  518 

22.0°  C.  548  mm.  533 

15.5°  C.  520  mm.  521 

36.0°  C.  567  mm.  558 

It  will  be  seen  by  a  careful  inspection  of  these  tables 
that  the  osmotic  pressure  was  found  to  be  proportional 
to  the  concentration,  and  to  vary  uniformly  with  the 
temperature.  The  variations  in  the  third  column  of 
the  first  table,  while  considerable  are  not  systematic, 
and  the  same  may  be  said  for  the  second  table  of  the 
differences  between  the  second  column  and  the  third, 
which  is  computed  by  the  formula 

P=  493(1  +  .003674 


202  KINETIC   THEORY. 

Similar  results  are  obtained  by  direct  observations  on 
other  substances.  Van't  Hoff1  concludes  from  these 
facts,  first  that  osmotic  pressures  follow  Boyle's  law, 
which  is  simply  one  way  of  stating  that  the  pressure, 
whether  gaseous  or  osmotic,  is  proportional  to  the 
concentration.  In  the  formula  given  above  the  tem- 
perature coefficient  is  the  same  as  that  for  ideal  gases, 
and  hence  he  concludes  that  osmotic  pressure  follows 
the  law  of  Gay-Lussac  and  Charles,  and  is  propor- 
tional to  the  absolute  temperature.  It  has  been 
shown  with  a  considerable  degree  of  exactness  that 
the  temperature  coefficient  of  the  osmotic  pressure  is 
the  same  for  solutions  of  different  substances  by  the 
following  method  :  If  a  protoplasmic  cell,  animal  or 
vegetable,  be  placed  in  a  solution  whose  osmotic 
pressure  is  greater  than  that  of  the  cell,  the  latter 
tends  to  shrink  and  shrivel ;  if  the  osmotic  pressure 
of  the  solution  be  less,  it  tends  to  swell.  This  action 
is  sufficiently  marked,  so  that  the  persistence  of  the 
cell  in  its  original  state  becomes  a  very  sensitive  test 
for  the  equality  of  the  osmotic  pressure  within  and 
without  the  cell.  It  has  been  observed  that  solutions 
of  diverse  substances  which  at  any  one  temperature 
have  the  same  osmotic  pressure  as  one  of  these  cells, 
are  also  at  any  other  temperature  in  equilibrium  with 
the  cell,  and  hence  with  each  other.  The  conclusion 
from  these  experiments  is  that  the  osmotic  pressures 
of  different  solutions,  including  the  cell -con tents,  have 
the  same  temperature  coefficient. 

^Ztschr.  Phys.  Chem.,  I.,  p.  481.    Harper's  Science  Series,  IV., p.  13. 


SOLUTIONS.  203 

To  establish  completely  the  relation  between  the 
laws  of  osmotic  pressure  and  of  gaseous  pressure,  it  is 
only  necessary  to  find  the  relation  between  the  con- 
stant factors  of  proportionality  for  each  substance  con- 
necting the  pressure  at  any  concentration  with  the 
temperature.  This  may  be  done  either  directly  or  in- 
directly. By  Avogadro's  rule  and  Boyle's  law  the 
pressure  of  an  ideal  gas  is  proportional  to  the  number 
of  molecules  in  unit  space.  Hydrogen  gas,  having  a 
density  at  o°  C.  and  760  mm.  pressure  of  .00009  gr- 
per  c.c.,  or  .09  gr.  per  liter  would,  if  present  to  the 
extent  of  2  gr.  per  liter  exert  a  pressure  equal  to 

760  x  2/.O9  =  16889  mm- 

If  there  were  such  a  thing  as  a  "sugar  gas,"  that  is, 
a  gas  composed  of  sugar  molecules  and  behaving  as 
an  ideal  gas,  this  would  exert  this  same  pressure  if 
present  to  the  amount  of  342  grams  per  liter,  this  be- 
ing the  molecular  weight  of  sugar,  and  if  present  to 
the  amount  of  10  grams  per  liter  would  exert  a  pres- 
sure of 

16889  x  IO/342  =  493  mm- 

Now  10  grs.  per  liter  is  very  nearly  the  amount  of 
sugar  present  in  a  I  per  cent,  solution,  and  493  is  ex- 
actly the  number  used  in  the  formula  giving  the  rela- 
tion of  the  osmotic  pressure  to  the  temperature  as  the 
osmotic  pressure  at  o°  C.  Hence  the  evidence  of  ex- 
periment is  that  the  sugar  in  solution  gives  an  osmotic 
pressure  very  nearly,  perhaps  exactly  equal,  to  that 


204  KINETIC   THEORY. 

which  would  be  exerted  at  the  same  temperature  by 
the  hypothetical  "  sugar  gas  "  having  the  same  num- 
ber of  molecules  in  the  same  space.  Experiment 
shows  that  this  relation  can  be  generalized,  and  that 
for  a  great  variety  of  substances  the  osmotic  pressure 
follows  this  law,  and  that  independent  of  the  solvents 
used.1  Van't  Hoff  states  the  relation  thus  :  "  Is-os- 
motic  solutions  contain  the  same  number  of  molecules 
of  the  dissolved  substance  in  the  same  volumes  at 
the  same  temperature  and  this  number  is  the  same 
that  would  be  contained  in  the  same  volume  of  an 
ideal  gas  at  the  same  temperature  and  pressure." 

Osmotic  pressure  may  also  be  studied  by  indirect 
methods.  The  osmotic  pressure  is  the  pressure  which 
must  be  exerted  to  prevent  more  of  the  solvent  from 
uniting  with  the  dissolved  substance  when  the  solution 
and  the  pure  solvent  are  separated  by  a  semi-per- 
meable membrane.  But  the  solvent  and  dissolved 
substance  may  be  separated  by  other  means  than  by 
forcing  the  solvent  through  a  membrane,  for  instance 
by  the  process  of  vaporization,  or  of  crystallization,  or 
by  presenting  another  solvent  which  will  dissolve  only 
one  of  the  components.  According  to  circumstances 
either  the  solvent  or  the  dissolved  substance  may  be  re- 
moved from  the  solution  by  either  of  the  three  methods. 
A  fairly  complete  discussion  of  them  all  is  given  by 
Nernst.2  We  shall  here  discuss  only  the  case  of  the 
removal  of  the  solvent  from  the  solution.  Suppose 

1  W.  C.  D.  Whetham,  Phil.  Mag.  (6),  5,  p.  282,  1903. 

2  "Theoretical  Chemistry,"  trans,  by  Palmer,  p.  124. 


SOLUTIONS. 


205 


the  solution  contained  in  a  receptacle  closed  at  the 
lower  end  by  a  semi-permeable  membrane,  and  con- 
tinued above  in  a  long  narrow  tube  open  at  the  top. 
Suppose  this  receptacle  set  in  a  jar  of  the  pure  solvent 
of  sufficient  depth  to  keep  the  semi-permeable  mem- 
brane covered  by  the  solvent,  and  further  imagine  the 
jar  tall  enough  to  enclose  the  long  tube,  and  her- 
metically sealed.  Then  the  solution  and  the  solvent 


Fig.  22. 

are  separated  in  different  places  by  the  walls  of  the 
receptacle,  by  the  semi-permeable  membrane,  and  by 
the  space  above  the  liquids  saturated  with  the  vapor 
of  the  solvent.  If  the  dissolved  substance  is  not 
volatile,  all  these  are  completely  impermeable  to  it, 
while  the  pure  solvent  is  able  to  pass  from  one  body 
of  liquid  to  the  other  either  through  the  membrane,  or 
through  the  vapor  by  the  process  of  vaporization  and 


206  KINETIC   THEORY. 

condensation.  When  a  state  of  equilibrium  is  attained 
the  upper  surface  of  the  solution  in  the  long  tube  will 
be  higher  than  the  surface  of  the  solvent  in  the  jar 
outside,  and  the  hydrostatic  pressure  due  to  the  differ- 
ence of  level  is  then  equal  to  the  osmotic  pressure. 
We  can  state  as  a  general  proposition  that  the  vapor 
tension  of  the  vapor  of  the  pure  solvent  over  the  solu- 
tion must  be  less  than  its  tension  over  the  solvent,  for 
if  it  were  not,  we  might  have  an  isothermal  cycle  in 
which  the  vapor  of  the  solvent  should  be  set  free  at 
the  surface  of  the  solution,  then  at  the  surface  of  the 
pure  solvent,  being  at  a  pressure  greater  than  that  at 
the  surface  of  the  solution  by  the  hydrostatic  pressure 
of  a  column  of  vapor  the  height  of  the  column  of  solu- 
tion, and  hence  at  a  pressure  greater  than  the  maxi- 
mum tension  over  the  pure  solvent,  it  would  condense, 
and  then  pass  through  the  semi-permeable  membrane 
into  the  solution,  a  cycle  which  would  take  place 
spontaneously  and  hence  offer  the  possibility  of  doing 
external  work.  The  denial  of  the  existence  of  such  a 
cycle  leads  to  the  declaration  that  when  a  state  of 
equilibrium  is  attained  the  difference  between  the  vapor 
pressures  on  the  surfaces  of  the  solution  and  of  the 
solvent  is  equal  to  the  hydrostatic  pressure  of  the 
column  of  vapor  equal  in  height  to  the  difference 
between  their  levels.  If  we  let 

h  =  difference  of  level  of  liquids, 

S  =  density  of  solution, 
/!  =  vapor  tension  over  solvent, 
pz  —  vapor  tension  over  solution, 


SOLUTIONS.  207 

M=  molecular  wt.  of  solvent, 
v  =  vol.  of  I  gr.  molecule, 
d  =  Mjv  =  mean  density  of  vapor, 

the  osmotic  pressure  will  be 

P=  hS, 

and  the  difference  of  vapor  tensions  will  be 

hM 

A-A  =  ^  =  —  • 

If  we  assume  that  the  vapor  behaves  as  an  ideal  gas 
the  equation 


enables  us  to  eliminate  v,  giving 

hpM 


and  eliminating  h 

(50)  P=-±-^.jfRT, 

where  /  is  a  mean  value  of  the  pressure,  intermediate 
between  /x  and  /2  and  hence  the  fractional  expression 
has  a  value  between  (pl  —  A) /A  anc*  (A  —  A) /A-  ^ 
more  rigorous  deduction  gives  its  value  as  log  p^p^ 
and  hence 

^  '  ~  M          °^  /2* 

Osmotic  pressure  can  be  determined  either  directly  by 
observing  the  depression  of  the  vapor  tension  and 


208  KINETIC   THEORY. 

computing  by  one  of  the  formulae  just  deduced,  or  by 
observing  the  elevation  of  the  boiling  point  or  depres- 
sion of  the  freezing  point  of  the  solvent  occasioned  by 
the  addition  of  the  dissolved  substance.  The  theory 
of  these  last  methods  is  intimately  related  to  that  of 
the  depression  of  the  vapor  tension,  and  their  formulae 
can  be  deduced  from  the  last  one  obtained.  In  Chap. 
V.  on  Change  of  State  we  have  deduced  the  formula 
for  the  latent  heat  of  vaporization, 


(29) 


vv  the  specific  volume  of  the  liquid  is  small  in  com- 
parison with  vv  the  specific  volume  of  the  vapor,  and 
if  we  neglect  the  former  and  assume  that  the  behavior 
of  the  latter  is  represented  nearly  enough  for  our  pur- 
poses by  the  equation  for  ideal  gases,  this  becomes 

L-T**** 

p  dT' 

dpLdT 


where  C  is  a  constant  of  integration.     If  TQ  be  the 
boiling  point  and  B  the  pressure  corresponding, 


SOLUTIONS.  209 

t> 
subtracting 


If  the  elevation  of  the  boiling  point  is  small  we  may 
call  it  t  and  obtain  the  equation 

pl  __  L    t 
'  A       R 

and  substituting  this  value 


Now  by  comparison  of  the  various  equations  it  appears 
that  L  was  the  amount  of  energy  in  mechanical  measure 
required  to  vaporize  one  gram  molecule  of  the  solvent, 
and  hence  L/M  is  the  energy  required  to  vaporize  one 
gram  and  differs  from  the  ordinary  value  of  the  latent 
heat,  X,  only  by  the  factor  J.  This  gives  us  the  form 

(52)  P=SJ\~. 

Jo 

An  interesting  special  form  is  that  for  the  osmotic 
pressure  of  a  substance  dissolved  in  water  which  is 


where  P  is  expressed  in  atmospheres. 

The  freezing  point  may  be  defined  as  the  tempera- 
ture at  which  the  solid,  liquid  and  vapor  phases  can 
coexist  in  equilibrium.  For  both  the  processes  of 
vaporization  and  of  sublimation  the  equation  for  latent 


210  KINETIC   THEORY. 

heat  which  we  wrote  a  little  while  ago  holds  good, 
hence  if  L  and  /  refer  to  vaporization  from  the  liquid 
and  Lr  and  p1  to  sublimation  from  the  solid, 

--+  C9 


log  /'=- 


If  the  freezing  point  be  TQ  and  the  vapor  tension  cor 
responding  be/0,  then 


Eliminating  Ct  Cf  and/0  by  successive  subtractions, 

p       L'-L(  i        i  \ 
-/-    -IT  \T-Tj- 

Now  in  this  expression  p  is  the  vapor  tension  over  the 
pure  solvent  at  the  temperature  7)  and  /'  is  the  ten- 
sion over  the  pure  ice  of  the  solvent  at  that  tempera- 
ture, and  hence,  from  the  condition  of  equilibrium, 
over  the  solution  ;  hence  we  may  substitute  this  value 
of  the  logarithm  in  the  equation  for  the  osmotic  pres- 
sure, which  becomes 

S(L'-L)t 


SOLUTIONS.  211 


and  since  (Lf  —  L)jM  is  the  amount  of  energy  re- 
quired to  melt  one  gram  of  the  solvent,  if  we  repre- 
sent the  ordinary  latent  heat  of  fusion  by  p  this 
becomes 


(53)  P 

0 

For  water  this  becomes,  in  atmospheres, 
P—  1  2.  07  A 

It  readily  appears  that  the  indirect  methods  of  deter- 
mining the  osmotic  pressure  are  not  capable  of  as 
great  accuracy  as  the  direct  method  might  give  if  we 
could  find  a  membrane  of  satisfactory  strength  and 
rapidity  of  action.  On  account  of  its  simplicity  and 
convenience  the  freezing  point  method  is  oftenest  used, 
and  as  often  only  an  approximate  value  of  the  osmotic 
pressure  is  desired  in  order  to  check  determinations 
of  molecular  weight,  its  accuracy  is  found  sufficient  to 
determine  which  of  two  or  more  otherwise  equally 
allowable  values  is  to  be  selected. 

We  have  then  three  equations  relating  the  osmotic 
pressure  to  the  depression  of  the  vapor  tension,  the 
elevation  of  the  boiling  point,  and  the  depression  of 
the  freezing  point,  respectively,  of  the  solvent  ; 


(50) 


(52)  P 

•*o 


212  KINETIC   THEORY. 

(53)  P=Sfy~. 

0 

We  have  also  the  law  experimentally  discovered  by 
Raoult l  that  the  relative  lowering  of  the  vapor  tension 
experienced  by  a  solvent  on  dissolving  a  foreign  sub- 
stance is  equal  to  the  ratio  of  the  number  of  dissolved 
molecules,  n,  to  the  number  of  molecules,  Ny  of  the 
solvent,  that  is 


~Y~  "N' 

p=jmRT-         ;'§| 

But  NM  is  the  number  of  grams  of  the  solvent  con- 
taining n  gram-molecules  of  the  dissolved  substance, 
or  NMjn  the  number  containing  I  gram-molecule, 
and  NM/nS  the  volume  of  the  solvent,  which  we  may 
call  V,  hence 


hence  in  a  dilute  solution,  for  which  Raoult' s  law 
holds,  and  for  which  the  approximations  made  are 
allowable,  as  for  instance  the  assumption  that  the 
volumes  of  the  solvent  and  solution  are  the  same, 
the  osmotic  pressure  follows  the  gas-laws.  This  is  an 
incidental  and  indirect  verification,  but  still  resting 
upon  experimental  evidence.  Other  evidence  of  about 
equal  weight  can  be  adduced  from  the  discussion  of 

1  Ztschr.  Phys.  Chem,,  2,  p.  353,  1888. 


SOLUTIONS.  213 

experiments  in  which  the  dissolved  substance  is  re- 
moved in  one  way  or  another  from  the  solution. 

Thermodynamics.  —  We  have  so  far  tacitly  assumed 
that  the  solutions  we  are  considering  are  such  that  if 
the  pure  solvent  be  added  to  them  it  will  diffuse  with- 
out any  resulting  change  of  temperature,  that  is,  that 
the  heat  of  dilution  is  zero.  It  may  be  shown  experi- 
mentally that  when  this  is  the  case,  the  osmotic  pres- 
sure is  proportional  to  the  absolute  temperature.  This 
may  be  shown  also  by  the  principles  of  thermody- 
namics. The  equation  for  the  first  law  of  thermo- 
dynamics may  be  written 


If  now  a  quantity  of  the  pure  solvent  and  of  the  solu- 
tion be  allowed  to  mix  freely  without  doing  any  work, 
and  without  applying  heat  or  cold,  we  have  dQ  and 
dW  both  vanishing,  and  hence 

SU  8U 


and  if  the  heat  of  dilution  be  zero,  there  will  be  no 
change  of  temperature,  that  is,  dT=  o  and  hence 


. 

~    dv  =  o  . 
dv 

But  if  by  v  we  mean  the  volume  accessible  to  the  dis- 


214  KINETIC   THEORY. 

solved  substance,  and  by  U  the  energy  of  its  molecules 

Ar+0, 

and  hence 

dU 

7V  =  °- 

The  second  law 

TdS  =  dU+  dW 

may  also  be  written 


which  may  be  separated,  since  T  and  v  are  entirely 
independent,  into  the  two  equations 


_ 

~ 


dT  ~  dT' 
8U 


dividing  by  T 


dT~  TdT 


differentiating  partially  by  v  and  T  respectively 

JjL    L^*L    ±(^L     ^\    JL^ 

&Tdv~~TdTdv~~T  *  \dvdT  '"*"  dT)      T\dv 


dT'~  T\  dv 
And  if 


SOLUTIONS.  215 


dp     p 


-~r-  =  const. 
•*(•) 

Or  in  words,  if  dUjdv  =  o,  then  when  the  volume  is 
kept  constant,  that  is,  the  concentration  kept  the 
same,  the  pressure  will  be  proportional  to  the  abso- 
lute temperature ;  but  this  condition  holds  true  for 
the  osmotic  pressure  if  the  heat  of  dilution  is  zero. 
Conversely  we  can  show  by  a  reversal  of  our  argu- 
ment that  if  the  osmotic  pressure  is  proportional  to 
the  absolute  temperature  the  heat  of  dilution  is  zero, 
and  this  latter  is  then  the  necessary  and  sufficient 
condition  for  the  former.  In  any  other  case  the  os- 
motic pressure  must  depart  from  the  simple  gas  laws. 


CHAPTER   X. 
KINETIC   THEORY   OF   SOLUTIONS. 

WE  shall  confine  ourselves  in  the  main  to  the  study 
of  solutions  in  which  the  dissolved  substance  is  non- 
volatile, the  vapor  consequently  consisting  of  mole- 
cules of  the  solvent  only.  Within  the  solution  we 
must  believe  that  the  molecules  both  of  the  dissolved 
substance  and  of  the  solvent  are  moving  freely,  and 
that  as  they  are  at  the  same  temperature  they  have 
the  same  mean  kinetic  energy  of  translation.  It 
seems  equally  certain  that  in  solutions  as  in  pure 
liquids  the  cohesive  or  intermolecular  forces  are  of 
large  amount.  The  phenomenon  which  we  have  to 
accept  as  fundamental  is  that  certain  films  allow  the 
passage  of  molecules  of  one  sort  and  not  of  the  other. 
Two  classes  of  these  films  are  observed,  the  solid 
semi-permeable  films,  which  may  be  animal  or  vege- 
table membranes,  or  membranes  of  precipitation,  and 
the  free  surface  of  the  liquid.  The  behavior  of  the 
first  class  of  membranes  is  comparatively  simple,  and 
whatever  may  be  the  mechanism  by  which  it  allows 
one  kind  of  molecule  to  pass  through  it,  while  stop- 
ping other  kinds,  the  fact  itself  is  sufficient  to  account 
in  general  for  the  phenomena  observed. 

The  free  surface  of  the  solution  allows  the  mole- 
cules of  the  solvent  to  pass  from  the  liquid  to  the 

216 


KINETIC  THEORY  OF  SOLUTIONS.  217 

vapor  regions  and  back  in  the  same  general  manner 
which  we  have  described  in  Chapter  VII.  on  Vaporiza- 
tion ;  the  passage  of  the  molecules  of  the  dissolved 
substance  through  this  surface  is  entirely  prevented. 
What  may  be  the  cause  of  this,  how  it  is  that  the 
molecular  forces  are  able  to  prevent  any  of  these 
molecules  from  penetrating  through  this  surface  we 
do  not  know.  For  present  purposes  however  we 
may  rest  satisfied  with  the  fact :  these  forces  probably 
act  through  a  region  whose  thickness  is  considerable 
as  compared  with  the  dimensions  of  molecules,  so 
that  there  is  in  this  region  a  gradual  diminution  in  the 
number  of  molecules  of  the  dissolved  substance  as 
one  passes  toward  the  free  surface.  We  can  then 
consider  that  in  the  main  body  of  the  solution  the 
molecules  of  the  dissolved  substance  are  uniformly 
distributed,  but  that  just  at  the  surface  there  is  a  film 
into  which  they  never  penetrate,  which  is  consequently 
composed  only  of  molecules  of  the  solvent,  this  film 
constituting  a  sort  of  buffer  between  the  solution  and 
the  vapor  over  it,  and  being,  if  you  choose,  the  semi- 
permeable  membrane. 

The  generalized  form  of  van  der  Waals*  equation 
we  have  written 

(45)  (p  +  P)(v-b)  =  RT. 

If  we  use  the  subscripts  a  and  b  to  refer  to  the  vapor 
and  liquid  states  respectively,  and  call  the  covolume 
v  —  b,  <E>,  this  becomes  for  the  solution,  in  which  the 
external  vapor  pressure  is  negligibly  small  as  com- 
pared with  the  molecular  pressure 


218  KINETIC   THEORY. 

(46)  Pt<kt-RT. 

This  equation  holds  equally  well,  with  the  change  of 
subscripts,  either  of  the  pure  solvent  or  of  the  solu- 
tion, the  value  of  R  being  the  same  for  quantities  of 
the  liquid  containing  the  same  number  of  molecules, 
since  for  the  same  temperature  the  mean  kinetic  ener- 
gies of  the  different  kinds  of  molecules  are  the  same, 
and 


If  we  let  the  subscript  b  refer  to  the  solution  and  w  to 
the  solvent,  which  for  convenience  we,shall  speak  of  as 
water,  then  for  equimolecular  quantities  of  the  two 
liquids 


The  pressure  Pb  within  the  solution  may  be  thought 
of  as  made  up  of  two  partial  pressures,  Pbw  due  to  the 
water  molecules,  and  Pba  due  to  the  molecules  of  the 
dissolved  substance.  To  determine  the  ratios  of  these 
exactly  would  require  a  careful  study  of  the  mean  free 
paths  of  two  kinds  of  molecules  in  a  mixture  where 
the  distances  between  the  molecules  are  of  the  order 
of  the  dimensions  of  the  molecules  ;  on  account  of  the 
exceeding  difficulty  of  this  determination  we  shall  con- 
tent ourselves,  as  a  first  approximation,  with  the  as- 
sumption that  these  partial  pressures  are  proportional 
to  the  numbers  of  molecules  which  occasion  them. 
We  shall  call  these  numbers  W  and  6*  respectively, 
and  the  whole  number  of  molecules  N.  Then 


KINETIC   THEORY   OF   SOLUTIONS.  219 

W+S=N 
and  the  partial  pressures  are 

W 
P          -  P 

•*&»  —    Jf^V 

P  -s-p 

bt      N   b' 

P    4-  P  —  P 
^iw  ^  1  bs  —  •rr 

In  Chapter  VII.  we  deduced  the  relation  between  the 
latent  heat  of  vaporization  of  the  pure  solvent  and  its 
pressures 

(48)       L,  -  RT  log      ±      +  (/>„  - 


which,  neglecting  in  each  case  the  smaller  pressure, 
becomes 


Following  the  reasoning  of  the  same  chapter,  if  N  be 
the  number  of  molecules  in  one  gram  molecule,  and 
<£6  the  covolume  of  one  gram  molecule  in  the  solu- 
tion, then  the  number  of  molecules  per  unit  covolume 
will  be  Nj$>h  and  the  number  of  these  striking  a  given 
unit  of  area  in  one  second  will  be  x 

1  Strictly,  the  number  is  the  sum  of  the  two  expressions 


W 

A.     __/  —  ~T     A. 


21/7T  *&   2I/7T 

but  as  only  the  first  term  is  used,  no  error  is  introduced  by  this  inaccu- 
racy of  form. 


220  KINETIC   THEORY. 

N 


But  if  we  consider  the  unit  area  as  taken  between  the 
homogeneous  solution  and  the  thin  surface  film  of 
solvent  which  the  dissolved  molecules  cannot  pene- 
trate, the  number  passing  this  unit  of  area  is  the  num- 
ber of  molecules  of  the  solvent  which  reach  it,  namely, 

W 


This  then  is  the  number  of  molecules  of  solvent  pass- 
ing up  from  the  solution  into  the  surface  film  of  pure 
solvent  per  unit  area  in  one  second.  This  surface  film 
is  also  the  non-homogeneous  layer  through  which  the 
vaporization  takes  place.  But  considering  a  thin  por- 
tion of  it,  next  to  the  solution,  so  thin  that  in  it  the 
covolume  can  be  considered  as  a  constant,  if  we  call 
this  covolume  <t>,  then  the  number  of  molecules  pass- 
ing down  from  this  portion  into  the  solution  through 
unit  area  in  one  second  will,  by  the  same  reasoning, 

be 

N    13 

$2i/7r* 
Since  these  numbers  must  be  the  same,  for  equilibrium, 


KINETIC   THEORY   OF  SOLUTIONS.  221 

That  is,  the  solution  is  covered  by  a  surface  film  of 
pure  solvent,  whose  lower  portions,  in  contact  with 
the  solution,  have  a  covolume  greater  than  that  of 
the  solution  in  the  proportion  of  N  to  W.  Then  the 
expression  for  the  latent  heat  of  vaporization  as  the 
work  of  the  molecules  in  passing  through  this  non- 
homogeneous  layer,  which  we  had  written 


becomes,  introducing  $  as  the  variable  of  integration, 
and  considering  that  the  molecule  has  to  pass  from  the 
lowest  part  of  the  surface  film  of  solvent  into  the  vapor 

^=Va-bw  Jfa 

Lb  =  RT\  (#.  +  *,)£. 

J  4>=N<t>b!W  "P 

which  gives  us,  if  we  assume  that  bw  is  a  constant, 

L  -- 
- 


which  by  comparison  with  the  equations 

Pd>  _  RT         p    .  -        p 
^b^b  —  ^  l  >        -^bw  —  ^y  *>> 

and  disregarding  small  quantities  as  before,  reduces  to 
the  form 


M      U 

which  is  entirely  analogous  to  the  expression  writteu 
for  a  pure  solvent,  substituting  for  the  molecular 
pressure  of  the  pure  solvent  its  partial  pressure  in  the 


222  KINETIC   THEORY. 

solution.  Subtracting  this  equation  from  the  other, 
and  replacing  Lw  —  Lb  by  La,  the  heat  of  dilution, 

L.  +  RT  \ogpf  =  RT  log  -'-  +  (/>„  -  PJtw  . 

Pb  ^bw 

This  expression  is  perfectly  general  ;  in  the  case  of 
dilute  solutions,  where  Pw  and  Pbw  are  very  nearly  the 
same  we  may  write 


=  *„(/>„  -  PJ, 

bw  *W 

and 

L,  +  RT  log  A  =(Pw-PJvu. 

•fb 

In  the  case  where   the  heat  of  dilution  is   zero,  this 
may  be  written 


Comparing  this  with  the  equation  deduced  in  the  last 
chapter, 

(50  />-£*riog£, 

we  see  that/w  =/t  is  the  vapor  tension  over  the  pure 
solvent,  pb  =  p2  is  the  tension  over  the  solution  and  vw  is 
the  volume  occupied  by  a  gram  molecule  of  this  solvent, 

MjSt  and  hence 

P—  P  —  P 

y  to  **** 

that  is,  the  osmotic  pressure  here  appears  as  the  differ- 
ence between  two  molecular  pressures.  More  explic- 
itly, if  a  solution  and  a  quantity  of  its  pure  solvent, 
both  under  the  same  external  pressure,  be  placed  in 


KINETIC   THEORY   OF   SOLUTIONS.  223 

communication  with  each  other  through  a  semi-per- 
meable membrane,  the  molecular  pressure  in  the  pure 
solvent  will  be  greater  than  the  partial  molecular  pres- 
sure of  the  solvent  in  the  solution,  and  hence  the  mole- 
cules of  the  solvent  will  tend  to  pass  into  the  solution, 
and  to  prevent  this  passage  it  is  necessary  to  apply  a 
hydrostatic  pressure  to  the  solution  equal  in  amount 
to  the  difference  between  these  molecular  pressures. 
This  hydrostatic  pressure  is  what  is  measured  in  the 
direct  determinations  of  osmotic  pressures. 

The  expression  Pw  —  Pbw,  which  we  have  just  found 
to  be  the  osmotic  pressure,  can  be  reduced  to  a  slightly 
different  form, 

P    P  -P    w P-P    N-SP 

rw       rbw  —  rw        jy  ^b  —  rw  jy     rv 

which  reduces,  if  Pb  =  Pw  to  the  form 

S 

N   "• 

That  is,  the  osmotic  pressure  is  proportional  to  the 
concentration  or  to  the  number  of  molecules  of  the 
dissolved  substance,  if  the  molecular  pressure  is  the 
same  both  for  the  pure  solvent  and  for  the  solution  ; 
in  this  case  also  the  covolume  will  be  the  same.  That 
this  last  condition  should  hold  rigidly  is  hardly  to  be 
expected,  yet  that  it  does  hold  approximately  is  shown 
by  the  experimental  discovery  of  Raoult's  law,  which 
states  the  same  relation  of  proportionality  to  the  num- 
ber of  molecules. 


CHAPTER   XL 

DISSOCIATION   AND   CONDENSATION. 

WHEN  the  atoms  or  radicals  which  go  to  make  up 
the  molecules  of  an  aggregate  are  capable  of  uniting 
in  different  combinations,  which  shall  result  in  different 
kinds  of  molecules,  and  are  also  capable  of  being  rear- 
ranged by  suitable  interchanges,  so  that  the  molecules 
of  certain  sorts  shall  be  made  to  increase  in  number  at 
the  expense  of  molecules  of  other  sorts,  our  experience 
as  formulated  in  the  teachings  of  physical  chemistry 
shows  that  such  interchanges  may  take  place  of  them- 
selves, without  the  intervention  of  external  controls. 
A  familiar  example,  which  may  serve  to  give  definite- 
ness  to  our  ideas  is  the  reaction  often  known  as  double 
decomposition,  such  as  the  reaction  of  sulphuric  acid 
and  common  salt  to  form  hydrochloric  acid  and  sodium 
sulphate,  in  accordance  with  the  formula 

H2S04  +  2NaCl  =  2HC1  +  Na2SO4. 

Still  simpler  are  the  reactions  classed  together  under 
the  name  of  dissociation,  of  which  a  striking  case  is  the 
dissociation  of  ammonium  chloride  on  vaporization,  in 
accordance  with  the  formula 

NH4C1  =  NH3  +  HC1. 

From  the  standpoint  of  chemical  dynamics  these  reac- 

224 


DISSOCIATION   AND    CONDENSATION.         225 

tions  are  reversible,  that  is,  may  take  place  in  either 
direction,  and  in  any  actual  case  will  probably  take 
place  in  both  directions,  with  speeds  depending  upon 
the  temperature  and  pressure  of  the  aggregate  and 
upon  the  concentration  of  the  various  kinds  of  mole- 
cules. According  to  these  views  equilibrium  is  attained 
simply  when  the  reactions  in  the  two  directions  are  of 
such  speed  as  to  leave  the  composition  of  the  aggregate 
unchanged. 

Kinetic  Theory  of  Dissociation.  —  Dissociation  of  a 
gas  which  results  in  a  change  of  the  number  of  mole- 
cules reveals  its  presence  by  anomalous  vapor  pressure, 
or  vapor  density,  according  to  the  circumstances  of  the 
experiment.  When  the  dissociation  is  complete,  we 
have  only  the  problem,  already  solved  at  least  approxi- 
mately, of  a  mixture  of  two  gases.  But  in  many  cases 
the  dissociation  is  only  partial,  and  is  found  to  depend 
either  upon  the  temperature  or  upon  the  density  of  the 
gas. 

One  possible  explanation  is  that  given  by  Boltzmann l 
and  in  slightly  different  forms  by  others.  According 
to  this  view,  whenever  two  atoms  or  radicals  which  are 
capable  of  uniting  with  each  other  come  sufficiently 
near  to  each  other,  and  in  suitable  relative  position, 
they  are  to  be  considered  as  forming  one  complex 
molecule.  The  problem  is  to  find  the  relative  num- 
ber of  such  pairs  of  atoms,  among  all  the  atoms  pres- 
ent, which  are  in  general  so  situated  relatively  as  to 
be  considered  as  chemically  combined,  under  the  exist- 

1  "  Gastheorie,"  II.,  pp.  177-217. 


226  KINETIC   THEORY. 

ing  conditions  of  mutual  attraction,  temperature,  pres- 
sure and  volume.  This  treatment  gives  the  degree 
of  dissociation  as  a  function  of  the  temperature  and 
pressure. 

Another  explanation,  different  in  form,  but  not  neces- 
sarily contradicting  the  first,  is  this  :  the  kinetic  energy 
of  any  molecule  consists  of  that  associated  with  its 
motion  of  translation,  and  that  associated  with  the 
relative  motions  of  its  parts.  This  latter  motion  has  a 
tendency  to  separate  the  parts  of  the  molecule,  so  that 
any  collision  between  two  molecules  which  is  so  con- 
ditioned as  to  increase  the  energy  of  the  internal  motions 
of  one  of  these  molecules  beyond  a  certain  amount  will 
result  in  its  actual  disruption.  Of  course  we  do  not 
know  that  there  is  any  simple  relation  between  the 
amount  of  internal  energy  of  an  individual  molecule 
and  its  external  energy.  But  we  have  come  to  believe 
that  for  any  large  body  of  gas  the  total  internal  energy 
has  a  definite  ratio  to  the  total  energy  of  translation  of 
the  molecules  (p.  73)  this  ratio  being  independent  of  the 
pressure  and  density,  and  probably  also  of  the  tempera- 
ture of  the  gas.  We  have  no  evidence  that  the  mole- 
cules attaining  the  highest  internal  energy  are  identically 
those  that  attain  the  highest  speeds,  but  it  is  reasonable 
and  necessary  to  believe  that  the  distribution  of  energies 
follows  the  same  laws.  In  the  same  way,  while  we  are 
not  able  to  say  anything  as  to  the  results  of  any  single 
collision  of  two  molecules,  since  the  relative  numbers 
of  molecules  having  the  different  speeds  is  constant,  in 
any  large  number  of  collisions,  the  resulting  speeds  of 


DISSOCIATION   AND   CONDENSATION.        22/ 

the  individual  molecules  must  have  the  same  distribu- 
tion, and  the  resulting  internal  energies  must  have  a 
distribution  entirely  similar  to  that  of  the  external  ener- 
gies of  translation. 

We  can  make  these  conceptions  more  definite  by 
applying  them  to  the  simplest  case  of  dissociation,  that 
in  which  the  molecule  is  dissociated  into  two  like  parts, 
as  in  the  case  of  iodine  vapor,  I2  =  2!  or  nitric  oxide, 
N2O4  =  2NO2.  If  the  whole  number  of  molecules, 
when  undissociated,  be  N,  we  shall  consider  that  part 
of  these,  Nt  molecules,  remain  undissociated,  while  the 
remainder,  N2  molecules,  dissociate  forming  2N2  of  the 
simpler  molecules,  so  that  the  resulting  gas  contains 
Nj  -j-  2N2  molecules,  while  the  original  gas  contained 
Nj  +  N2  =  N  molecules. 

Suppose  that  the  critical  internal  energy  just  capable 
of  producing  dissociation  corresponds  to  the  speed  cl  then 
the  number  of  collisions  in  any  given  time  resulting  in 
such  dissociation  will  be  proportional  to  the  number  of 
collisions  of  the  undissociated  molecules,  NVP1  =  -A^  ?//t 
(p.  60)  and  to  the  probability  of  speeds  above  the  criti- 
cal speed  fv 


(p.  25)  that  is,  to  the  product 


Writing  x  for  c/a,  and  applying  formulae  of  integra- 


228  KINETIC   THEORY. 

tion  developed  on  pp.  27  and  31, 


T 


in  which  all  the  terms  of  the  series  except  the  first 
two  may  ordinarily  be  neglected,  since  ^  is  to  be  re- 
garded as  very  large  in  comparison  with  a,  and  hence 
x^  =  c^a.  is  very  large.1 

Since  the  dissociated  and  undissociated  parts  of  the 
gas,  being  thoroughly  mixed,  are  at  the  same  temper- 
ature, and  hence  have  the  same  average  kinetic  energy 
per  individual  particle,  and  since  the  former  have  only 
half  the  mass  of  the  latter,  their  speed  will  be  greater 
in  the  ratio  1/2  :  I  so  that  the  number  of  collisions  of 
one  of  these  dissociated  molecules  taking  place  with 
its  speed  less  than  a  certain  critical  speed  c2  would  be 


r 
Vo      l/2 


where  P2,  the  number  of  collisions  per  second  of  a  dis- 
sociated particle,  can  be  replaced  by  i/2<r//2  and  we 
may  write  ;r=  <:/T/2a,  giving 

1  Jaeger,  in  Winkelmann's  "  Handbuch,"  II.,  2,  pp.  563-4,  assumes 
that  c  can  be  substituted  for  c,  while  in  this  integration  c  >  ^  ^>  c  so 
that  the  result  here  found  ought  to  be  more  accurate. 


DISSOCIATION   AND   CONDENSATION.        229 


4v  2c  ra 

VirL  Jo 


The  probability  that  the  other  party  to  the  collision 
shall  have  a  similarly  low  speed  is,  using  y  instead  of  x, 


so  that  the  number  of  collisions  between  pairs  of  such 
molecules  is  represented  by  the  product  of  these  two 
quantities  by  2NV  the  number  of  such  particles,  and 
divided  by  2,  since  each  collision  involves  two  of  them, 
giving 


rv^v  r% 

Jo  Jo 


Since  the  speeds  of  different  molecules  are  entirely  in- 
dependent, we  may  multiply  these  expressions  under 
the  radical  sign,  giving 


and  remembering  that  the  only  requisite  is  that  the 
total  energy  of  the  two  particles  shall  not  exceed  a 
given  limit,  that  is  that  x*  -f  j2  shall  not  exceed  a  given 
value,  say  r2,  we  can  let  x  =  r  cos  6,  y  =  r  sin  0  and 
integrate  between  proper  limits,  giving 

v  cos2  ^  sin2  OrdrdO. 


7T/2 

The  expression  to  be  integrated  becomes,  on  separating 
the  variables 


230  KINETIC   THEORY. 


fV*Wr  r/2cos2  6  si 


2  = 


sn 


which  can  be  integrated  by  successive  applications  of 
formula  (8)  on  p.  27,  giving 


so  that  our  original  expression  becomes,  writing  ^2  for  r, 


In  reviewing  the  development  of  these  expressions 
we  need  to  remember  that  the  speeds  c  and  the  ratios 
x  are  used  merely  as  a  means  of  determining  the  num- 
bers of  collisions  having  certain  properties,  while  the 
real  point  at  issue  is  whether  the  internal  energy  of  a 
given  molecule  exceeds  (or  falls  below)  a  certain  limit. 
The  difference  in  treatment  of  the  two  cases  arises  from 
the  fact  that  in  the  first  case  we  consider  the  result  to 
one  molecule  only,  while  in  the  second  we  are  inter- 
ested in  a  result  involving  two,  namely  that  the  two 
colliding  particles  shall  have  the  sum  of  their  energies 
less  than  a  certain  amount,  and  that  they  shall  actually 
combine  and  remain  together  for  a  time. 

Whenever  a  steady  state  of  dissociation  is  attained, 
not  only  will  the  temperature  of  the  dissociated  and 
undissociated  portions  be  the  same,  but  the  number  of 
molecules  dissociating  and  the  number  of  collisions 
resulting  in  reassociation  of  molecules  will  be  the  same, 
that  is, 


DISSOCIATION   AND   CONDENSATION.         231 


[I  -K*W +«••'+*)]. 


in  which  c  has  the  same  meaning  on  both  sides,  l^  and 
/2  depend  upon  the  dimensions  of  the  different  kinds 
of  molecules,  and  xl  and  .r2  are  ratios  depending  upon 
the  temperature  of  the  gas,  and  upon  the  requisite  ener- 
gies at  which  the  dissociation  and  reassociation  take 
place,  or  in  other  words,  the  temperatures  of  dissocia- 
tion and  condensation. 

We  have  not  yet  taken  account  of  the  effect  of  vary- 
ing density  upon  the  degree  of  dissociation.  Experi- 
ment indicates  that  dissociation  is  less  as  the  density  is 
greater,  or  in  other  words,  is  greater  as  the  volume, 
and  the  mean  free  path,  increase.  This  may  be  due 
to  the  greater  number  of  triple  or  multiple  collisions, 
or  of  collisions  following  so  closely  as  to  be  regarded 
as  multiple  in  the  more  dense  gas.  We  may  take 
account  of  this  by  multiplying  the  first  member  of  the 
equation  just  deduced  by  a^  in  which  a  is  simply  a 
factor  of  proportionality.  Doing  this,  and  introducing 
another  value  of  /2,  namely 

I 

T/27T7Z0-2 ' 

in  which 

n  =  -^> 

and  dividing  by  2c 


232  KINETIC   THEORY. 


aNjr9*  /  i 


[I- 


_  -  j^W  +  2*.'  +  2)] 

'"'     "  " 


which  may  also  be  written 


If  the  gas  be  sufficiently  rarefied  to  follow  the  laws 
of  ideal  gases,  aside  from  the  dissociation,  its  behavior 
can  be  represented  by  the  equation 


where  k  is  a  suitable  constant,  and  a.  the  coefficient  of 
expansion.  Multiplying  together  the  appropriate  mem- 
bers of  the  last  two  equations,  and  dividing  by 


fit)          N*       N. 


N.         .  -  N.      N.      N 

- 


or    in    other   words,  the    proportion    of  the    original 


DISSOCIATION   AND   CONDENSATION.        233 
molecules  which  are  dissociated  is 


N     vT+V 

where  z  is  defined  as  a  function  of/  and  /  by  two  pre- 
ceding equations. 

From  this  last  result, 


-, 

i/T+V 


L=\ 
i  +  zJ 


and  the  law  of  the  gas  is 

pv  =  kN(  I  +  _L=Y(I  +  at\ 

\          Vi+zJ 

differing  from  that  of  the  ideal  gas  only  in  the  factor 
involving  z.  If  the  density  of  an  ideal  gas  at  a  given 
pressure  and  temperature  be  called  d^  that  of  a  disso- 
ciated gas  at  the  same  temperature  and  pressure  d,  these 
densities  will  be  inversely  as  the  volumes,  and  hence 


I  +• 


For  numerical  computations  it  is  necessary  to 
evaluate  x^  and  ;r2  as  functions  of  the  temperature. 
In  the  first  case,  that  of  double  molecules,  the  definition 
was 


a*      aa\l  +  at)' 


234  KINETIC   THEORY. 

where  the  first  a  is  the  most  probable  speed  of  the 
molecule,  and  the  subscript  o  refers  to  values  at  the 
temperature  o°C.  If  c*  be  regarded  as  the  "mean 
square,"  then 


In  the  second  case, 

c         **- 


and  the  value  of  x*  for  each  of  two  particles  having  the 
same  speed,  and  having  the  maximum  value  of  the 
sum  of  their  energies  compatible  with  reassociation 
would  be 

where 
and  hence 


-  2(1  +  at)  ' 

which  enable  us  to  express  /(/)  and  hence  z  as  func- 
tions of  t,  tv  and  t2. 

The  assumption  that  x2  is  very  small  while  x^  is  very 
large  leads  to  the  following  simplifications ;  in  the 
numerator  of f(t)  we  can  expand  the  exponential  into  a 
series 


hence,  keeping  only  the  first  term  of  the  result, 


DISSOCIATION   AND   CONDENSATION.        235 
I  -  K**W  +  ix?  +  2) 

=  i  -  «i  -  *?  +  K4  -  K6)(2  +  2*2  +  *i4)  =  K6 

and  similarly  rejecting  1/2^  in  comparison  with  ^ 


(!  +  «*)«' 

where  B  and  ft  are  new  constants,  and 

pB     2* 


+at)     '    k  (i  +at)l 


a  result  which  differs  only  very  slightly  from 


obtained  by  Jaeger,  on  slightly  different  assumptions. 
He  gives  the  following  tabulation  of  the  density  of 
nitric  oxide,1  in  which  the  computed  values  are  found  by 

i  Winkelmann,  "  Handbuch  der  Physik,"  II.,  2,  p.  568.     The  table 
is  based  upon  data  given  in  A.  Naumann,  "  Thermochemie, "  p.  177. 


236 


KINETIC   THEORY. 


substituting  the  values  ^=3.18,  A 
y8  =  23.83,  in  the  last  equation, 


1501  x  10 


-11 


t 

^(obs.) 

d  (comp.) 

t 

</(obs.) 

d  (comp.) 

26.7° 

35-4° 
39-8° 
49.6° 
60.2° 
70.0° 

2.65 

2.53 
2.46 

2.27 
2.08 
1.92 

2.70 

2-55 
2.46 
2.27 
2.07 
1.92 

80.6° 
90.0° 
100.1° 
111.3° 
121.5° 
135-0° 

.80 
•72 

.68 

•65 
.62 
.60 

•79 
.72 
.67 
.64 
.62 
.60 

The  value  of  /3  here  employed  enables  us  to  give  an 
approximate  value  to  tv  the  dissociation  temperature, 

£=f(i  +00=23.83, 
1+^=2.23.83, 

/!  =  4064°. 

There  is  reason  for  believing  that  changes  in  the 
degree  of  association,  that  is,  polymerization  and  dis- 
sociation, occur  very  largely  in  the  liquid  and  solid 
states,  and  in  connection  with  the  change  from  one 
state  to  another.  Two  results  follow  immediately  ; 
the  introduction  of  further  corrections  into  our  equation 
of  condition,  and  a  decided  increase  in  the  apparent 
values  of  the  specific  heat  and  the  latent  heat  of  change 
of  state.  It  is  interesting  to  note  that  beginning  with 
the  type  equation 

Nine* 


and  introducing  the  correction  factor  due  to  the  dissocia- 
tion, by  simple  approximations,  based  on  the  assump- 


DISSOCIATION   AND    CONDENSATION.         237 

tion  that  v  is  large  compared  with  b,  either  van  der 
Waal's  equation  or  that  of  Clausius  may  be  deduced.1 
In  this  connection  Sutherland  in  his  paper  on  "  The 
Molecular  Constitution  of  Water"2  has  brought  for- 
ward the  view  that  while  steam  has  its  molecule  cor- 
rectly represented  by  the  ordinary  formula  H2O,  ice  is 
really  (H2O)3,  and  water  a  solution  of  (H2O)3  in  (H2O)2; 
on  this  basis  he  has  developed  formulae  which  account 
quantitatively  for  the  behavior  of  water,  often  anoma- 
lous, in  nearly  every  particular.  The  triple  formula 
for  ice  suggests  an  equilateral  triangle  as  the  dominant 
feature  of  the  form  of  the  complex  molecule,  which  is 
consistent  with  the  persistence  of  the  angle  60°  in  its 
crystals.  He  finds  that  the  assumption  of  a  density 
of  .88  for  liquid  (H2O)3  (trihydrol}  and  1 .089  for  (H2O)2 
{dihydrof}  at  o°  C.,  and  proportions  varying  from  37-5 
per  cent,  by  weight  of  trihydrol  at  o°  to  21.7  per  cent, 
at  1 00°,  and  probably  nearly  pure  dihydrol  at  the  crit- 
ical temperature,  will  account  for  the  maximum  density 
at  4°  C.,  for  the  diminution  of  the  optical  coefficient 
(n2  —  i)/(«2  -f  2)p  with  rising  temperature,  for  the  pecu- 
liarities of  the  compressibility  of  water,  and  for  its  char- 
acteristic surface  tension  and  viscosity.  In  general, 
either  a  rise  of  temperature,  or  an  increase  of  pressure, 
has  a  tendency  to  dissociate  trihydrol  into  dihydrol,  so 
that  the  surface  film,  in  which  there  is  a  tension,  rather 
than  a  pressure,  is  almost  pure  trihydrol  at  tempera- 

1  Winkelmann,  "Handbuch  der  Physik,"  II.,  2,  p.  569. 

*  Phil.  Mag.  (5),  50,  pp.  460-489,  1900.  Nernst  ("Theoretical 
Chemistry,"  p.  650)  states  that  work  on  surface  tension  indicates  a 
degree  of  association  for  water  varying  from  2.3  to  3.$, 


238  KINETIC   THEORY. 

tures  below  40°,  and  is  richer  in  this  ingredient  at 
higher  temperatures  than  is  the  body  of  the  liquid. 
The  large  values  of  the  latent  heats  of  fusion  and  vapor- 
ization are  easily  accounted  for  by  the  superposition  of 
latent  heats  of  dissociation  ;  in  particular,  in  the  case 
of  fusion,  the  shrinkage  might  otherwise  even  seem  to 
call  for  a  "  latent  cold "  of  fusion.  For  numerical 
results,  reference  is  made  to  the  original  paper. 

Electrolytic  Dissociation. — Van't  HofT,  in  an  early 
paper  on  Osmotic  Pressure,1  noted  that  while  most 
substances  in  dilute  solution  exhibit  osmotic  pressures 
very  nearly  agreeing  with  those  deduced  from  the  gas- 
laws,  there  are  exceptions  which  systematically  show 
pressures  considerably  greater  than  these,  so  that  to 
include  all  cases  the  equation  must  be  written 

pv  =  iRt, 

where  i  may  have  values  greater  than  I.  The  lim- 
iting value,  I,  applies  to  the  non-exceptional  cases. 
Soon  afterwards  Svante  Arrhenius,2  a  Swedish  chemist, 
suggested  a  possible  method  of  explaining  these  excep- 
tions. In  the  case  of  gases  anomalous  densities  or 
pressures  are  commonly  explained  on  the  basis  of  dis- 
sociation. Arrhenius  suggested  that  it  was  only  natural 
to  explain  these  anomalous  osmotic  pressures  in  the 
same  way  ;  Van't  Hoff  had  noticed  that  the  exceptions 
included  all  the  salts,  all  the  acids,  all  the  alkalies, 
that  is,  all  the  substances  which  in  solution  conduct 

1  Ztschr.  phys.  Chem.,  I.,  481,  Harper's  Science  Series,  IV.,  p.  13. 
*Ibid.,  L,  63 1;  Harper's  Science  Series,  IV.,  p.  47. 


DISSOCIATION   AND    CONDENSATION.        239 

electrolytically.  Arrhenius  takes  up  a  suggestion  made 
by  Clausius 1  in  1857  that  a  part  of  the  molecules  of  a 
solution  which  conducts  electrolytically  are  dissociated, 
and  that  the  conduction  is  by  means  of  these  dissociated 
or  active  molecules.  But  he  goes  further  than  Clausius 
by  making  this  idea  quantitative,  and  not  merely  qual- 
itative. The  conductivity  of  a  solution  will  be  propor- 
tional to  the  number  of  these  " active"  molecules  or 
ions,  and  hence  from  measurements  of  the  conductivity 
one  can  determine  the  "  coefficient  of  activity,"  a.  The 
method  of  the  determination  can  be  taken  up  later. 
Now  this  coefficient  a.  is  intimately  related  with  the 
coefficient  i  of  Van't  HofTs  equation.  If  a  solution 
contain  molecules  n  in  number  before  their  dissociation, 
and  the  coefficient  of  activity  be  a,  then  na  will  be  the 
number  of  the  original  molecules  which  have  suffered 
dissociation,  and  n(\  —  a)  the  number  remaining  undis- 
sociated ;  if  each  molecule  on  dissociation  forms  k 
parts,  then  after  the  dissociation  instead  of  n  molecules 
there  will  be 

n(  I  —  a)  -f  nak  =  n  [  I  -J-  a(k  —  I )]  , 

and  the  osmotic  pressure  will  be  increased  in  the  same 

ratio,  hence 

z  =  I  -f.  a(k  —  l). 

A  first  immediate  consequence  of  this  theory  is  that 
when  the  electrolyte  is  largely  dissociated,  any  prop- 
erties of  the  solution  which  are  due  immediately  to  the 
ions  themselves  should  be  additive.  This  may  be 

iPogg.  Ann.,  101,  p.  347. 


240  KINETIC   THEORY. 

tested  in  general  by  comparing  the  property  in  ques- 
tion for  the  salts  of  a  given  ion  with  the  corresponding 
salts  of  some  other  ion.  Thus  we  are  familiar  with  the 
characteristic  blue  of  dilute  solutions  of  cupric  solu- 
tions, the  orange  of  bichromates,  the  deep  purple  of 
permanganates  ;  Ostwald l  for  thirteen  salts  of  perman- 
ganic acid  has  measured  the  position  in  the  spectrum 
of  four  principal  absorption  bands,  and  found  them 
identical ;  so  that  a  colored  ion  gives  its  color  to  the 
solution  regardless  of  the  presence  of  another  ion,  and 
if  this  second  ion  be  colorless,  the  effect  of  the  colored 
one  is  immediately  evident.  In  fact  the  use  of  indica- 
tors to  determine  the  acidity  or  alkalinity  of  solutions 
seems  to  depend  upon  a  difference  in  color  between  an 
undissociated  molecule  and  one  of  its  ions. 

Valson 2  showed  that  the  specific  gravity  of  salt  solu- 
tions was  an  additive  property,  that  is,  the  difference 
between  the  specific  gravities  of  equimolecular  (dilute) 
solutions  of  salts  of  two  given  different  metals  with  the 
same  acid  was  constant,  independent  of  the  acid  and 
similarly  for  the  salts  of  two  given  acids  with  the  same 
base.  These  conclusions  have  been  corroborated  by  the 
later  work  of  Ostwald  on  the  change  of  volume  occur- 
ring in  the  case  of  the  neutralization  of  acids  with  bases. 

Similarly,  the  molecular  refractive  constant  (either  in 

Mn—i)        M(n2—i) 

the  form  -     —j—  -  or  -5^3-7  —  {  in  which  M  is  the 
d  d(n*  +  2) 

^Ztschr.  phys.  Chem.,  9,  584. 
*£/?.,  73»  P-  44i,  1874,  etc. 


DISSOCIATION   AND    CONDENSATION.         241 

molecular  weight,  and  d  the  density)  and  the  optical 
activity,  that  is  the  power  of  rotating  the  plane  of 
polarized  light,  are  additive  properties  of  the  ions. 

Even  more  convincing  are  the  results  of  the  study 
of  electrical  conductivity,  of  mixed  solutions,  and  of  the 
heat  of  neutralization  of  strong  acids  and  bases.  It  is 
a  common-place  that  mixing  equimolecular  weights  of 
two  salts  of  strong  acids  and  strong  bases,  as  sodium 
chloride,  and  potassium  bromide,  at  considerable  dilu- 
tion, the  properties  of  the  mixture  are  indistinguishable 
from  those  of  a  similar  mixture  of  sodium  bromide  and 
potassium  chloride.  This  is  an  obvious  necessity,  if 
both  salts  in  each  case  are  completely  dissociated,  for 
the  two  solutions  then  contain  identical  mixtures  of  the 
four  ions,  sodium,  potassium,  bromine,  chlorine.  But 
a  mixture  of  methyl  chloride  and  ethyl  bromide  is  en- 
tirely distinct  in  its  properties  from  one  of  ethyl  chloride 
and  methyl  bromide,  none  of  these  compounds  being 
dissociated  in  solution. 

It  will  appear  later  that  water  itself  is  practically 
undissociated,  hence  if  a  completely  dissociated  acid 
solution  be  added  to  a  completed  dissociated  alkaline 
solution,  and  the  resulting  salt  be  soluble  and  also 
completely  dissociated,  the  only  chemical  action  which 
will  take  place  is  the  union  of  the  hydrogen  and 
hydroxyl  radicals,  and  the  heat  developed  will  be  inde- 
pendent of  the  kind  of  acid  and  alkali  employed.  That 
this  is  true  in  the  case  of  strong  acids  and  bases  is 
shown  by  the  following  table  taken  from  Nernst.1 

1  "Theoretical  Chemistry,"  p.  510. 
16 


242  KINETIC    THEORY. 

TABLE  OF  THE  HEATS  OF  NEUTRALIZATION  OF  ACIDS  AND  BASES. 

Acid  and  Base.  Heat  of  Neutralization. 

Hydrochloric  acid  and  sodium  hydroxide,  13,700 

Hydrobromic  acid  and  sodium  hydroxide,  13,700 

Nitric  acid  and  sodium  hydroxide,  13,700 

lodic  acid  and  sodium  hydroxide,  13,800 

Hydrochloric  acid  and  lithium  hydroxide,  I3,7OO 

Hydrochloric  acid  and  potassium  hydroxide,  13,700 

Hydrochloric  acid  and  barium  hydroxide,  13,800 

Hydrochloric  acid  and  calcium  hydroxide,  I3,9OO 

The  difference  between  the  two  types  of  dissociation 
is  well  illustrated  by  ammonium  chloride,  which  on 
being  vaporized  breaks  up  according  to  the  equation 

NH4C1  =  NH3  +  HC1, 

but  in  aqueous  solution  is  dissociated  into  the  ions 
NH4C1  =  NH4  +  Ci. 

Electrolytic  dissociation  then  differs  in  most  marked 
fashion  from  the  gaseous  dissociation  which  we  have 
previously  discussed  and  from  the  similar  dissociation 
and  polymerization  in  solution,  in  that  the  resulting 
ions  carry  electric  charges,  and  that  these  charges  are 
all  equal  in  amount  or  simple  multiples  of  the  unit 
ionic  charge.  This  latter  peculiarity  is  shown  by  Far- 
aday's law,  that  the  amount  of  material  going  into  or 
out  of  the  solution  at  either  electrode  is  proportional 
to  the  electrical  current  and  the  time,  that  is,  to  the 
quantity  of  electricity  involved,  and  also  to  the  com- 
bining weight  of  the  ion  ;  or  in  other  words,  the  same 
current  sets  free  chemically  equivalent  quantities  of  any 


DISSOCIATION   AND   CONDENSATION.        243 

ions  in  the  same  time.  Ions  are  univalent,  divalent, 
trivalent,  according  as  they  carry  once,  twice,  or  three 
times  the  unit  ionic  charge.  This  definition  is  in  agree- 
ment with  ordinary  chemical  usage. 

Whenever  the  ions  find  themselves  in  an  electrostatic 
field  due  to  the  presence  of  charged  electrodes  in  the 
solution,  they  experience  forces  tending  to  make  the 
positive  ions  move  from  the  positively  charged  elec- 
trode toward  the  negative,  the  negative  ions  in  the 
opposite  direction.  The  resulting  drift  of  charged  ions 
is  then  the  mechanism  of  the  electrolytic  conduction 
of  the  electric  current.  It  is  customary  to  call  the 
positively  charged  electrode  the  anode,  the  other  the 
cathode,  while  the  positively  charged  ion,  from  its  ten- 
dency to  be  liberated  at  the  cathode,  is  called  the 
cation,  the  negative  the  anion.  Cations  include  in  gen- 
eral metallic  and  basic  atoms  and  radicals,  and  the 
replaceable  hydrogen  of  acids,  while  anions  include 
acid  radicals  and  hydroxyl. 

Anions  and  cations  do  not  necessarily  travel  with  the 
same  speed.  In  the  extreme  case,  if  one  set  did  not 
move  at  all,  the  current  would  consist  entirely  of  the 
carriage  of  charges  of  one  sign  by  the  ions  of  that 
kind,  but  in  the  ordinary  case,  positive  charges  are 
being  carried  away  from  the  anode  toward  the  cathode, 
and  negative  charges  in  the  opposite  direction,  the 
whole  current  being  thus  made  up  of  the  sum  of  these 
two  effects. 

If  the  ions,  on  arriving  at  the  electrodes,  go  out  of 
the  solution,  there  is  evidently  an  impoverishment  of 


244  KINETIC   THEORY. 

the  solution  at  those  points,  for  at  an  electrode  the  ions 
which  arrive  correspond  to  a  part,  only,  of  the  current; 
while  those  going  out  of  the  solution  at  that  point  cor- 
respond to  the  whole  current  ;  while  if  the  electrode 
is  dissolved  by  a  secondary  reaction  regenerating  the 
solution,  there  is  a  corresponding  concentration  of  the 
solution  in  that  region,  the  new  ions,  and  the  newly 
arrived  ions,  more  than  making  up  for  those  which 
have  migrated  toward  the  other  electrode.  In  either 
case,  by  suitably  dividing  the  solution  after  the  current 
has  passed  for  some  time,  and  determining  by  chemical 
analysis  the  impoverishment  or  concentration  of  the 
solution  in  the  neighborhood  of  the  electrodes  it  is 
possible  to  determine  the  ratio  of  the  speeds  of  migra- 
tion of  the  two  ions. 

Since  the  passage  of  the  current  is  a  matter  of  the 
carriage  of  charges  of  both  kinds  by  the  ions,  the 
conductivity  of  a  solution  is,  other  things  being  equal, 
proportional  to  the  sum  of  the  speeds  of  migration  of 
the  ions.  Conductivity  data,  therefore,  together  with 
the  data  on  the  ratios  of  speeds  of  the  molecules,  en- 
able us  to  determine  quantities  u  and  v  proportional  to 
the  ionic  speeds,  such  that 


v), 


where  /*  is  the  molecular  conductivity,  that  is,  the 
actual  conductivity  divided  by  the  concentration,  and 
a  the  ionization  constant  or  coefficient  of  activity.  It 
then  appears  that  the  values  of  u  or  v  for  the  same  ion 


DISSOCIATION   AND   CONDENSATION.        245 

in  the  same  solvent  are  the  same,  independent  of  the 
other  ion  making  up  the  salt,  so  that  from  a  table  of 
values  of  u  and  v  for  various  ions,  the  conductivities 
of  solutions  of  any  salts  formed  from  pairs  of  these 
ions  can  be  prophesied,  if  only  the  degree  of  dissocia- 
tion can  be  foretold.  This  result  is  due  to  Kohlrausch. 
It  is  evidently  possible  from  such  data  as  we  have 
mentioned  to  deduce  the  values  of  the  absolute  veloci- 
ties of  the  ions.  According  to  Kohlrausch  the  abso- 
lute velocities  under  a  potential  gradient  of  one  volt 
per  cm.  is  found  by  multiplying  the  relative  velocities 
u  and  v,  by  the  factor  1 10  x  icr7.  He  gives  the  ab- 
solute velocity  of  the  hydrogen  ion,  under  these  condi- 
tions as  .0032  cm.  per  second.  Lodge1  filled  a  long 
U  tube  with  a  solution  of  gelatine  and  sodium  chloride, 
colored  with  phenolphthalein  with  a  trace  of  sodium 
hydroxide  to  bring  out  the  red  color.  This  solution 
hardened  sufficiently  to  prevent  any  convective  mixing 
but  did  not  seem  to  affect  perceptibly  either  the  true 
diffusion,  or  the  migration  of  the  ions.  The  tube  was 
inverted  into  two  vessels  containing  electrodes  im- 
mersed in  dilute  acid.  The  rate  of  progress  of  the 
hydrogen  ions  was  shown  by  the  retreat  of  the  color 
in  the  gelatine  tube.  The  result,  corrected  for  the 
natural  rate  of  diffusion  of  the  acid,  is  the  rate  of  ad- 
vance of  the  hydrogen  ions  due  to  the  current.  He 
found,  for  unit  gradient,  the  values  .0029,  .0026, 
.0024.  These  results,  which  would  naturally  be 
slightly  below  the  true  value,  correspond  satisfactor- 

1  B.  A,  Report,  1886,  p.  393. 


246  KINETIC   THEORY. 

ily  with  that  computed  by  Kohlrausch.  Wetham1 
has  measured  the  velocity  of  certain  colored  ions, 
observing  a  level  surface  between  two  solutions  of 
different  density.  His  results  also  confirm  the  theory. 
To  drive  one  gram  of  hydrogen  ions  through  water  at 
the  rate  of  I  cm.  per  second,  the  force  required  has 
been  computed  to  be  equal  to  about  320,000  tons 
weight. 

If  we  imagine  a  rectangular  electrolytic  cell2  of 
which  two  of  the  parallel  surfaces  are  of  platinum  I 
cm.  apart,  the  height  of  the  cell  being  indefinite,  intro- 
ducing a  liter  of  water  containing  in  solution  a  gram 
molecule  of  the  salt  to  be  investigated,  (e.  g.,  58.5  g.  of 
common  salt)  and  measuring  the  resistance  between  the 
platinum  faces  used  as  electrodes,  the  reciprocal  of  this 
resistance  represents  the  molecular  conductivity,  that 
is,  the  conductivity  due  to  the  ions  produced  in  that 
solution  from  a  gram- molecule  of  salt.  If  we  add 
more  water,  making  in  all  two  liters,  and  again  deter- 
mine the  conductivity,  we  shall  find  it  increased.  In- 
creasing dilution  will  still  increase  the  molecular  con- 
ductivity, but  only  up  to  a  certain  limit,  which  is 
reached  in  the  case  of  this  salt  at  a  dilution  of  about 
10,000  liters.  The  explanation  is  that  the  dissociation 
is  at  first  incomplete,  perhaps  two  thirds  of  the  mole- 
cules being  dissociated  in  the  normal  solution,  becom- 
ing complete  only  at  great  dilutions.  The  limiting 
value  is  often  called  the  molecular  conductivity  at  in- 

1  Phil.  Trans.,  1893,  A,  p.  337. 

2  Walker,  "Int.  to  Phys.  Chem.,"  pp.  220-221. 


DISSOCIATION   AND   CONDENSATION.        247 

finite  dilution,  and  indicated  by  the  symbol  /iw.     Then 
the  degree  of  dissociation  is  represented  by 


while  it  is  evident  that 

pn  =  u  -f-  v. 

This  latter  relation  evidently  gives  a  method  of  obtain- 
ing /-I*,  for  compounds  for  which  it  cannot  be  found  by 
direct  experiment,  if  both,  or  even  one  of  its  ions  occur 
in  other  compounds  which  can  be  completely  disso- 
ciated. 

For  the  compounds  which  are  even  at  great  dilution 
only  partly  dissociated,  sometimes  called  half-electro- 
lytes, Ostwald  has  found  that  the  degree  of  dissociation 
is  related  to  the  degree  of  dilution  by  the  equation 


where  k  is  often  called  the  dissociation  constant.  This 
equation  was  deduced  on  theoretical  grounds  from  the 
law  of  mass  action,  a  law  which  was  really  assumed  in 
the  deduction  of  the  law  of  dissociation  of  gases,  in  the 
earlier  part  of  this  chapter.  A  physical  meaning  is 
given  to  k  by  making  a.  =  i,  when 


-  l>      l>  — 

(\          —    /v.  /P     — — 

i-.5> 


248  KINETIC   THEORY. 

or  in  words,  k  is  the  reciprocal  of  twice  the  volume  of 
dilution  necessary  to  secure  the  dissociation  of  one  half 
the  molecules  of  the  electrolyte.  In  the  case  of  acetic 
acid,  for  which  /£=. 000018,  ijv  =  . 000036;  or  a 
solution  of  acetic  acid  .oooO36th  normal  would  only  be 
one  half  dissociated. 

The  strongly  dissociated  compounds  do  not  follow 
this  law,  but  other  dilution  formulas  have  been  devised, 

as  that  of  Rudolphi, 

a? 
— —  =  Const, 

or  that  of  van't  Hoff 

rt\ 

=  Const., 


which  agree  fairly  well  with  observed  facts,  but  are 
purely  empirical. 

Conductivity  depends  not  only  on  the  degree  of  dis- 
sociation, which  affects  the  number  of  ions  available, 
but  also  upon  the  size  of  the  ions  and  the  viscosity  of 
the  solution,  which  affect  their  speed.  The  lessened 
viscosity  accounts  for  the  increase  of  conductivity  in  a 
conducting  solution  as  the  temperature  is  raised.  That 
this  is  so  is  shown  both  by  the  fact  that  such  increase 
of  temperature  may  be  shown  experimentally  to  affect 
the  degree  of  ionization  only  slightly,  and  by  the  effect 
of  adding  small  quantities  of  alcohol  or  glycerine,  which 
are  known  to  increase  the  viscosity  of  water  very 
greatly,  but  not  to  affect  the  ionization,  with  a  great 
resulting  increase  of  the  electrolytic  resistance. 


DISSOCIATION  AND   CONDENSATION.        249 

The  degree  of  dissociation  of  an  electrolyte  in  solu- 
tion is  affected  by  the  presence  of  other  dissolved  sub- 
stances, and  particularly  by  the  addition  of  a  second 
electrolyte  containing  one  ion  in  common.  In  this 
case  the  dissociation  is  made  measurably  less.  Such 
an  effect  is  shown  qualitatively  by  passing  HC1  gas 
into  a  saturated  solution  of  NaCl.  The  excess  of  Cl 
ions  causes  the  recombination  of  some  of  the  previously 
dissociated  NaCl,  which  supersaturates  the  solution 
with  those  molecules,  resulting  in  their  precipitation. 
Noyes l  has  used  the  change  of  solubility  of  the  only 
slightly  soluble  thallous  chloride  to  determine  the 
degree  of  dissociation  of  other  chlorides  in  solution, 
obtaining  results  consistent  with  those  obtained  by  the 
other  methods. 

It  is  noticeable  that  the  so-called  strong  acids  or 
bases  are  precisely  those  which  are  most  strongly  dis- 
sociated in  solution,  and  that  electrolytes  as  a  class 
enter  into  chemical  reactions  promptly  and  vigorously. 
In  fact,  the  strength  of  the  strong  bases  and  acids 
seems  to  be  due  primarily  to  their  dissociation,  their 
radicals  being  thus  free  from  "  entangling  alliances  " 
and  ready  to  enter  into  any  possible  combination  in 
much  greater  numbers  than  is  possible  to  the  radicals 
of  a  less  completely  dissociated  compound.  In  this 
connection,  it  is  notable  that  almost  all  chemical  reac- 
tions take  place  either  in  aqueous  solutions  or  at  least 
in  the  presence  of  traces  of  moisture ;  so  that  thor- 
oughly dried  chlorine  gas  seems  to  have  no  effect  on 

1  Ztschr.  phys.  Chem.,  9,  603  ;   12,  162  ;   13,  412  ;   16,  125. 


250  KINETIC    THEORY. 

fused  sodium,  and  either  no  or  very  slight  action  on 
most  other  metals ;  in  dry  oxygen  dried  charcoal  will 
burn,  but  without  flame,  forming  both  carbon  monoxide 
and  carbon  dioxide,  while  sulphur,  boron  and  phos- 
phorus do  not  burn  at  all ;  dry  acid  does  not  affect 
litmus  ;  dry  hydrochloric  acid  does  not  form  a  precipi- 
tate when  passed  through  silver  nitrate  dissolved  in 
ether  or  benzol,  nor  act  on  dry  ammonia.  These  in- 
stances could  easily  be  multiplied,  but  they  serve  to 
call  attention  to  the  part  played  by  water  and  ioniza- 
tion  in  rapid  chemical  actions.  It  is  true,  however, 
that  many  reactions  are  known,  some  very  rapid, 
which  do  not  seem  to  be  in  any  way  dependent  upon 
the  presence  of  water,  or  upon  dissociation  into  elec- 
trically charged  ions. 

Special  interest  centers  in  the  theory  of  electrolytic 
cells  used  as  batteries  for  the  production  of  electric 
currents.  The  thermodynamical  theory  of  reversible 
cells  has  been  given  by  Helmholtz.  Imagine  a  small 
cycle,  which  may  be  considered  as  a  Carnot's  cycle, 
in  the  first  part  of  which  the  battery  is  allowed  to  gen- 
erate a  current,  working  isothermally  at  the  tempera- 
ture T  until  it  has  delivered  a  unit  quantity  of  elec- 
tricity. Then  the  amount  of  electrical  energy  developed 
will  be  numerically  equal  to  E,  the  E.M.F.  of  the  cell, 
while  chemical  reactions  will  have  taken  place  which, 
but  for  their  electrical  utilization,  would  have  produced 
the  quantity  of  heat  q  ;  hence  the  amount  of  heat  which 
must  be  supplied  from  without  to  keep  the  tempera- 
ture constant  is  the  difference  E  —  q.  If  now  the 


DISSOCIATION   AND   CONDENSATION.        251 

temperature  be  lowered  to  T  —  dT,  and  the  direction 
of  the  current  reversed  while  unit  quantity  of  electricity 
is  passed,  the  E.M.F.  and  hence  the  electrical  energy 
absorbed,  will  be  E  —  dE,  while  the  heat  value  of  the 
chemical  reaction  will  be  q  —  dq.  Returning  the  cell 
to  its  original  temperature,  it  will  also  be  in  its  original 
state  electrically  and  chemically,  while  the  residue  of 
electrical  work  done  by  it  is  dE.  The  efficiency  of  the 
cell  as  a  heat  engine  must  be 

dE     _dT 
E—q"T* 

whose  solution  is 


In  most  reversible  cells  the  first  term,  the  heat  energy 
of  the  chemical  reactions,  is  the  most  important,  but 
the  complete  theory  has  been  conclusively  verified 
experimentally  by  Jahn  ]  and  others. 

Nernst2  has  developed  a  theory  which  relates  the 
phenomena  of  the  voltaic  cell  to  those  of  osmotic  pres- 
sure. If  two  portions  of  the  same  solvent  are  brought 
into  contact  with  each  other,  one  portion  containing  a 
given  electrolyte  in  solution,  the  other  not  containing 
it,  diffusion  will  begin  to  take  place  immediately.  Now 
as  a  rule  the  two  ions  of  an  electrolyte  do  not  have  the 
same  mobility,  and  hence  will  not  be  able  to  diffuse 

1  Wied.  Ann.,  28,  pp.  21  and  491,  1886. 

*Ztschr.  phys.  Chem.,  4,  129,  1889.  "Theoretical  Chemistry," 
pp.  607-616. 


252  KINETIC   THEORY. 

into  the  new  space  at  the  same  rate,  but  the  more 
mobile  one  will  be  present  in  greater  numbers  in  the 
region  of  less  concentration,  giving  it  an  electrification 
of  its  own  sign,  while  the  region  of  greater  concentra- 
tion will  have  an  excess  of  the  less  mobile  ions,  and  a 
charge  of  the  corresponding  sign.  The  result  will  be 
a  difference  of  potential,  which  may  be  made  available 
as  an  electromotive  force  in  a  so-called  concentration 
cell.  This  electromotive  force  then  appears  imme- 
diately as  a  phenomenon  of  the  varying  osmotic  pres- 
sures of  the  two  ions  in  the  different  parts  of  the 
solution. 

Whenever  any  soluble  material  is  in  contact  with  its 
solvent,  its  tendency  to  go  into  solution  can  be  stated 
in  terms  of  its  solution  pressure,  this  quantity  being 
analogous  to  osmotic  pressure,  and  measured  in  the 
same  units,  so  that  the  substance  dissolves,  is  in  equi- 
librium, or  is  precipitated  from  the  solution,  according 
as  the  osmotic  pressure  is  less  than,  equal  to,  or 
greater  than  this  solution  pressure.  This  conception 
can  be  applied  equally  well  to  simple  solution  of  inert 
substances,  or  to  the  solution  of  a  metallic  electrode, 
whose  atoms  pass  into  the  liquid  as  positively  charged 
ions.  When  a  metal  is  immersed  in  a  solvent  which 
is  not  saturated  with  its  ions,  there  is  immediately  a 
rush  of  these  ions  into  the  solution  ;  then  the  solution 
pressure  of  the  metal  may  be  held  in  equilibrium  by 
the  opposing  electrostatic  field,  due  to  the  positive 
charges  of  the  ions  and  the  resulting  equal  negative 
charge  of  the  metal,  as  in  the  case  of  silver  dipped 


DISSOCIATION   AND   CONDENSATION.         253 

into  a  solution  of  salt,  NaCl,  or  copper  in  dilute  sul- 
phuric acid ;  if  the  solution  pressure  is  sufficiently 
high,  the  electric  forces  developed  may  be  so  great  as 
to  drive  out  of  the  solution  other  positive  ions,  as  when 
iron  is  dipped  into  copper  sulphate  solution,  and  copper 
is  precipitated  upon  the  iron  in  quantity  equivalent, 
electrically,  to  the  iron  dissolved. 

An  illustration  showing  the  harmony  of  the  two 
methods  of  discussions  is  the  copper  sulphate  concen- 
tration cell.  Suppose  two  copper  electrodes  dipping 
one  into  a  region  of  low,  the  other  into  a  region  of 
high  concentration.  Then  about  the  former  there 
is  less  osmotic  pressure  of  copper  ions,  hence  more 
tendency  for  the  copper  to  go  into  solution,  resulting 
in  a  current,  if  the  external  circuit  be  completed, 
which  within  the  liquid  will  pass  from  the  less  con- 
centrated to  the  more  concentrated  portion  of  the 
cell.  This  will  increase  the  concentration  of  the  posi- 
tive copper  ions  in  the  region  of  less  concentration,  and 
lessen  it  about  the  other  electrode.  The  more  rapid 
diffusion  of  the  negative  sulphion  ions  into  the  region 
of  less  concentration  tends  to  favor  the  current  in  the 
same  direction,  so  that  the  net  result  within  the  cell  is 
a  tendency  to  uniformity  of  concentration,  by  diffusion, 
by  migration  of  sulphion  ions,  and  by  the  passing  of 
copper  ions  into  or  out  of  the  solution  at  the  electrodes. 

Cells  of  the  Daniell  type  are  of  especial  interest.  They 
may  be  represented  by  the  symbols 

Zn,     ZnSO4,     CuSO4,     Cu. 


254  KINETIC   THEORY. 

The  solution  pressure  of  Zn  is  much  greater  than  that 
of  Cu,  so  that  when  the  external  circuit  is  closed  more 
Zn  ions  pass  into  the  solution  while  an  equal  number 
of  Cu  ions  pass  out  onto  the  copper  electrode,  with  a 
corresponding  current  having  the  direction  from  zinc 
to  copper  within  the  cell.  It  readily  appears  that  the 
effect  of  diluting  the  solution  of  zinc  sulphate  about 
the  zinc  is  to  increase  the  ease  with  which  the  zinc 
goes  into  solution,  and  hence  the  E.M.F.  of  the  cell, 
while  diluting  the  copper  sulphate  solution  for  the  same 
reason  will  lower  the  E.M.F. 

The  analogy  between  vaporization  and  solution 
makes  it  possible  to  give  exact  mathematical  form  to 
this  theory.  If  the  osmotic  pressure  follows  Boyle's 
law,  then  the  energy  of  the  isothermal  transformation 
from  the  osmotic  pressure/  to  the  solution  pressure  P 
of  the  metal,  that  is,  the  energy  available  from  the 
solution  of  the  metal,  and  hence  the  difference  of 
potential  between  metal  and  electrolyte  is 


and  neglecting  the  difference  of  potential  between  the 
two  solutions,  the  E.M.F.  of  the  Daniell  cell  amounts 
to 


where  the  subscript  I  refers  to  zinc,  and  2  to  copper. 
This  result  implies  that  the   E.M.F.   of  the  cell  is 


DISSOCIATION   AND   CONDENSATION.         255 

dependent  only  upon  the  cation,  and  not  upon  the 
anion.  This  is  found  to  be  substantially  true  in  the 
case  of  solutions  which  are  of  the  same  concentration 
and  ionization,  unless,  as  in  the  case  of  the  NO3  ion, 
there  is  action  between  the  electrode  and  the  solution 
not  considered  in  the  deduction. 

This  theory  enables  us  to  localize  the  E.M.F.  of  the 
battery  cell,  as  existing  mainly  between  the  electrodes 
and  the  surrounding  solution,  and  from  measurements 
of  the  potentials  there  developed  to  compute  the  solu- 
tion pressure,  P  for  the  various  metals.  These  range l 
from  io44  atmospheres  for  magnesium  and  io18  for  zinc 
to  io"20  for  copper.  These  solution  pressures  seem  to 
be  constants,  dependent  upon  the  solvent,  as  well  as 
the  metal,  and  upon  the  temperature,  but  independent, 
in  general,  of  the  anion. 

In  the  irreversible  type  of  cell  the  anode  and  its 
phenomena  are  similar  to  those  of  the  reversible  type 
of  cell,  but  the  cation  which  would  otherwise  be  set 
free  at  the  cathode  is  either  absorbed  by  a  block  of 
carbon  or  oxidized;  in  the  former  case  the  E.M.F.  is 
considerably  reduced,  the  cell  is  said  to  be  polarized ; 
in  the  latter  the  oxidation  furnishes  a  considerable 
supply  of  energy,  and  often  cations  of  a  different  char- 
acter, usually  materially  increasing  the  E.M.F.  of  the 
cell,  which  is,  however,  usually  very  inconstant. 

The  phenomena  of  electrolytic  dissociation  are  con- 
fined to  solutions  in  a  very  limited  number  of  solvents, 
and  are  altogether  most  noticeable  in  water  solutions. 

1  Nernst,  " Theoretical  Chemistry,"  p.  410. 


256  KINETIC   THEORY. 

These  solvents  are  all  anomalous  in  other  respects, 
giving  evidence  of  molecular  complexity,  and  ex- 
hibiting very  unusually  high  dielectric  constants,  that 
of  water  being  the  very  highest.  Attention  has  been 
called  by  J.  J.  Thomson 1  and  Nernst 2  to  the  influence 
which  the  dielectric  constant  must  have  upon  the  elec- 
tric forces  existing  between  the  ions,  these  forces  being 
inversely  as  the  dielectric  constant ;  so  that  in  a  sol- 
vent having  a  veiy  large  dielectric  constant  the  separa- 
tion into  ions  is  opposed  by  much  smaller  electrical 
forces.  The  high  degree  of  polymerization  of  these 
solvents  may  be  a  cause  of  their  high  dielectric  con- 
stants. No  satisfactory  quantitative  relation  has  yet 
been  discovered  between  the  dielectric  constants  of  the 
different  solvents  and  their  ionizing  power,  although 
the  following  statement  will  illustrate  the  qualitative 
relation  : 

Water,  dielectric  constant  about  80,  ionizing  power 
greatest. 

Formic  acid,  dielectric  constant  62,  ionizing  power 
about  three  fourths  that  of  water. 

Methyl  alcohol,  dielectric  constant  about  33,  ionizing 
power  from  one  half  to  two  thirds  that  of  water. 

Ethyl  alcohol,  dielectric  constant  about  26,  ionizing 
power  from  one  fourth  to  one  third  that  of  water. 

While  water  is  the  strongest  dissociating  solvent,  its 
own  tendency  is  toward  polymerization  rather  than  dis- 
sociation, and  conductivity  measurements  have  shown 

*  Phil.  Mag.  (5),  36,  p.  320,  1893. 

*  Ztschr.  phys.  Chem.,  13,  p.  531,  1894. 


DISSOCIATION   AND    CONDENSATION.        257 

so  marked  a  dependence  of  that  conducting  power 
upon  the  presence  of  impurities  that  some  have  been 
ready  to  declare  that  pure  water  is  an  absolute  non- 
conductor. The  evidence  seems  to  indicate  that  water 
itself  is  actually,  though  only  very  slightly,  dissociated. 
Determinations  by  different  methods  give  concordant 
results,1  the  concentration  C0  of  the  ions  in  gram-ions 
per  liter  being  at  25°  C.: 

By  conductivity  ^Q  —    •&  x  IO~6  (KohlrauscJi). 

By  electromotive  force  CQ=  i.o  X  io~7  (Ostwald). 

By  hydrolysis  C0  —  I .  I  x  I  O~7  (Arrhenius- 

Shields). 

By  saponification          £0=  I-2  X  IO~7  (Wijs). 
Kohlrausch  and  Heydweiller  ( 1 894)  give 

C,=    -3373  X  icr7^  18°. 

While  there  are  some  gaps  in  the  theory  of  elec- 
trolytic dissociation,  and  some  discrepancies  to  be  ex- 
plained, a  sane  review  of  the  facts  in  evidence  seems 
to  indicate  that  the  main  points  of  the  theory  are  well 
established,  and  the  discrepancies  are  such  as  to  lead 
to  the  development  rather  than  the  overthrow  of  the 
doctrine.2 

lonization  of  Gases,  —  The  application  of  a  sufficient 
potential -difference  will  cause  an  electric  spark  to  pass 

'Nernst,  "Theoretical  Chemistry,"  p.  662. 

2  See  Wetham,  Phil.  Mag.  (6),  5,  pp.  279-290,  1903. 

17 


258  KINETIC   THEORY. 

between  conducting  terminals  in  air,  while  the  passage 
of  the  current  can  be  maintained,  once  started,  by  a 
much  smaller  potential.  The  passage  of  the  current 
is  much  easier  through  the  moderately  rarefied  air  of  a 
Geissler  tube,  and  in  the  high  vacuum  of  a  Crookes 
tube  assumes  quite  a  different  character,  the  glow  pro- 
ceeding in  straight  lines  from  the  cathode  regardless 
of  the  position  of  the  anode.  It  further  appears  that 
flames  or  even  heated  gases  show  considerable  con- 
ducting power,  that  rarefied  gases  through  which  an 
electric  discharge  is  passing  are  excellent  conductors 
of  currents  due  to  small  potentials  also,  as  in  the 
Zehnder  tube l  for  the  detection  of  electric  waves  ;  and 
that  air  exposed  to  ultra-violet  light,  to  Roentgen  rays, 
or  the  radiations  from  radium  salts  and  other  radio- 
active substances  has  lost  its  insulating  properties  and 
acquired  a  considerable  conducting  power.  To  explain 
these  facts  one  of  the  theories  early  advanced  was  that 
of  an  electrolytic  dissociation 2  of  the  gas  ;  the  greatest 
development  has  come  with  the  study  of  the  effects  of 
radiation.  J.  J.  Thomson  and  Rutherford 3  have  shown 
that  air  which  has  been  exposed  to  Roentgen  rays 
exhibits  a  behavior  entirely  analogous  to  that  of  a  very 
dilute  electrolyte.  In  their  apparatus,  air  after  being 
subjected  to  the  radiation  was  caused  to  pass  through 
an  earth-connected  metal  tube,  in  whose  axis  is  placed 

1  Wied.  Ann.,  47,  pp.  77-92,  1892. 

2Giese,  Wied.  Ann.,  17,  p.  538,  1882,  etc.     Schuster,  Proc.  Roy. 
Soc.,  p.  317,  1884.    Elster  and  Geitel,  Wied.  Ann.,  37,  p.  324,  1889. 
il.  Mag.  (5),  42,  pp.  392-407,  1896. 


DISSOCIATION   AND   CONDENSATION.        259 

a  wire  charged  to  a  high  potential.  They  found  that 
with  a  steady  stream  of  gas  passing  through  the  tube, 
an  increase  of  potential  produced  a  corresponding  in- 
crease in  the  leakage-current  only  up  to  a  certain  limit, 
when  the  current  became  "saturated."  The  explana- 
tion is  that  the  current  depends  both  upon  the  potential 
and  upon  the  supply  of  ions  available,  so  that  if  the 
supply  is  veiy  limited,  the  maximum  current  is  that 
which  will  just  exhaust  the  supply  of  ions.  This  view 
is  enforced  both  by  the  fact  that  the  air  thus  exhausted 
is  no  longer  a  conductor  and  by  the  other  fact  that 
when  the  leakage  takes  place  between  parallel  plates, 
if  the  potential  is  high  enough  to  saturate  the  current, 
the  current  increases  continually  with  increase  of  the 
distance  between  the  plates,  the  latter  ranging  in  the 
experiments  reported  from  .  I  to  8  mm.,  this  apparent 
violation  of  Ohm's  law  being  due  to  the  increase  in 
the  number  of  ions  available  for  the  conduction  of  the 
current ;  but  with  smaller  potential  difference,  the  cur- 
rent at  first  increased,  then  decreased  as  the  distance 
became  greater  between  the  electrodes. 

It  is  possible  that  the  hot  gases  from  flames  which 
have  been  fed  with  suitable  salts  contain  ions  similar  to 
those  present  in  aqueous  solution,  which  may  furnish 
the  mechanism  for  the  conduction  of  the  electric  cur- 
rent, and  may  also  be  concerned  in  giving  the  charac- 
teristic spectrum  of  the  vapor.  But  in  many  cases  the 
ions  concerned  are  of  a  different  type,  called  by  J.  J. 
Thomson  corpuscles,1  which  seem  to  be  smaller  than 

*  Phil.  Mag.  (5),  44,  P-  3",  1897. 


260  KINETIC   THEORY. 

ordinary  atoms,  and  of  a  nature  independent  of  the 
source  from  which  they  are  derived ;  so  that  the  ca- 
thode rays  seem  to  be  a  stream  of  negatively  charged 
corpuscles,  driven  off  by  electrostatic  repulsion,  moving 
in  straight  lines  unless  deflected  by  a  magnetic  or  an 
electrostatic  field,  and  causing  phosphorescence  wher- 
ever it  strikes  the  glass. 

The  velocity  of  these  corpuscles  was  found  to  be,  in 
Thomson's  experiments,  of  the  order  of  10°  cm.  per 
sec.,  depending  upon  the  difference  of  potential  be- 
tween the  electrodes,  as  compared  with  io5  for  ordinary 
gas  molecules,  while  the  mean  effective  velocity  under 
unit  potential-gradient  of  the  ions  in  air  which  has  been 
exposed  to  Roentgen  radiation,  as  measured  by  the 
conducting  power,  is  1.6  cm.  per  sec.,1  or  for  hy- 
drogen 5.2  cm.,  as  compared  with  .003  cm.  per  sec.  for 
the  hydrogen  ion  in  aqueous  solution. 

The  ratio  of  the  mass  of  the  ion  to  the  charge  which 
it  carries  is,  in  the  case  of  the  electrolysis  of  liquids,  the 
common  electrochemical  equivalent.  This  ratio  has 
been  determined  by  Thomson  in  the  case  of  the 
cathode  rays  from  a  study  of  their  deviation  by  a 
magnetic  or  electrostatic  field,  since  the  curvature  of 
the  path  depends  upon  the  balance  between  the  "  cen- 
trifugal force,"  due  to  their  inertia  and  velocity,  and 
the  deviating  force,  dependent  in  the  electrostatic  field 
on  the  charges,  in  the  magnetic,  both  on  charge  and 
velocity.  He  finds  that  this  ratio2  is  very  nearly  icr7 

*  Phil.  Mag.  (5),  44,  p.  434,  1897. 
*PM.  Mag.  (5),  44,  p.  310,  1897. 


DISSOCIATION   AND   CONDENSATION.        261 

as  compared  with  the  smallest  previous  known  value, 
io~4,  for  the  hydrogen  ion  (expressed  in  the  same  units). 
Later  work  indicates1  that  the  charges  carried  by 
these  corpuscles  are  of  the  same  magnitude  as  those 
carried  by  ions  in  liquids,  so  that  the  corpuscles  them- 
selves are  shown  to  be  exceedingly  small  in  compar- 
ison with  ordinary  molecules  and  atoms.  This  deter- 
mination was  made  possible  by  the  discovery  of  C.  T. 
R.  Wilson2  that  dust-free  moist  air,  which  has  been 
ionized  by  Roentgen  radiation,  will  produce  a  cloud  on 
being  subjected  to  an  expansion  which  would  not  pro- 
duce the  cloud  in  un-ionized  air.  The  air  in  question 
was  kept  saturated  with  moisture,  and  the  expansion 
was  of  a  measured  amount,  so  that  the  quantity  of 
water  condensed  could  be  computed,  while  the  rate  at 
which  the  cloud  settled  through  the  air  enabled  the 
computation  of  the  size  of  the  individual  drops,  and 
hence  their  number.  The  fact  that  these  corpuscles, 
though  vastly  smaller  than  the  smallest  atoms,  can 
serve  as  nuclei  for  this  condensation,  a  function  ordi- 
narily performed  only  by  solid  dust  particles,  is  ex- 
plained by  the  statement3  that  on  a  charged  sphere  of 
less  than  a  certain  radius,  the  effect  of  the  charge  in 
promoting  condensation  will  more  than  counterbalance 
the  effect  of  surface  tension  in  preventing  it.  As  a  con- 
sequence, these  charged  ions,  in  spite  of  their  diminu- 


il.  Mag.  (5),  46,  p.  544,  1898  ;   (6),  5,  p.  354,  1903. 
2  Phil.  Trans.  A.,  1897,  p.  265. 

3J.  J.  Thomson,  "Applications  of  Dynamics  to  Physics  and  Chem- 
istry," p.  164. 


262  KINETIC   THEORY. 

tive  size,  collect  minute  drops  of  water  which  act  as 
centers  of  condensation. 

It  appears  then  that  these  corpuscles  are  very  minute 
objects,  so  small  that  it  takes  hundreds  or  thousands 
of  them  to  make  ordinary  atoms,  but  carrying  unit 
ionic  charges  (usually  negative).  Being  so  small, 
their  collisions  with  molecules  must  be  regarded  as 
really  collisions  with  the  corpuscles  constituting  these 
molecules,  so  the  number  of  collisions  of  a  single  cor- 
puscle and  hence  its  free  path,  depends  upon  the  total 
number  of  corpuscles  present,  including  those  in  the 
molecules,  and  hence  simply  upon  the  density  of  the 
gas,  and  not  on  its  material  (as  corpuscles  from  all  dif- 
ferent sources  are  indistinguishable  in  their  properties). 

These  corpuscles  are  undoubtedly  present  to  some 
extent  in  any  gas,1  being  formed  and  destroyed  by 
recombination  continuously.  When  a  region  contain- 
ing such  gas  is  subjected  to  a  very  strong  electric  field, 
these  corpuscles  are  given  very  high  velocities  in  the 
direction  of  the  field,  so  high  that  striking  against  gas 
atoms  they  are  able  to  dissociate  or  ionize  them,  thus 
increasing  the  number  of  such  carriers,  so  that  they 
are  adequate  for  the  conveyance  of  a  considerable  cur- 
rent. This  seems  to  be  the  beginning  of  the  passage 
of  an  electric  spark.  The  ionization  by  collisions  can 
begin  only  when  the  field  is  strong  enough  to  give  the 
ions  an  energy  or  velocity  which  exceeds  a  certain  crit- 
ical value,  sufficient  to  ionize  a  molecule  by  collision, 
and  reaches  its  limit  when  the  conductivity  is  so  great 

*  Phil.  Mag.  (5),  50,  pp.  278-283,  1900. 


DISSOCIATION   AND   CONDENSATION.        263 

as  to  reduce  the  strength  of  the  field  to  or  below  this 
limit.  But  the  energy  given  to  a  corpuscle  of  charge 
e  by  a  field  of  strength  F  in  traveling  the  distance  /,  its 
mean  free  path,  is  evidently  measured  by  the  product 
Fel,  so  that  the  field  F  required  to  produce  a  spark 
varies  inversely  as  /,  and  hence  directly  as  the  density 
of  the  gas,  a  result  which  we  know  to  be  approxi- 
mately true.  We  can  see  also  why  there  might  be 
great  difficulty  in  securing  a  discharge  between  elec- 
trodes so  near  that  the  existing  ions  would  have  very 
little  opportunity  for  hitting  and  ionizing  the  gas  mole- 
cules. 

Corpuscles  are  thus  produced  in  considerable  num- 
bers by  the  collisions  of  previously  existing  corpuscles 
acting  in  a  strong  electric  field  ;  they  are  also  produced 
by  the  action  of  certain  "  radiations,"  some  of  which, 
as  the  Roentgen  rays  and  ultra-violet  light,  seem  to  be 
phenomena  of  the  ether,  others,  as  the  cathode  and 
Becquerel  rays,  seem  to  be  streams  of  corpuscles,  while 
the  radiations  from  other  substances  seem  to  consist  of 
both  types.  They  are  also  given  off  by  heated  bodies, 
which  suggests  to  J.  J.  Thomson  *  a  possible  explan- 
ation of  some  of  the  phenomena  of  the  solar  corona, 
and  of  comets. 

The  corpuscles  disappear  from  a  gas  either  by  spon- 
taneous recombination,  in  which  case  the  number  pres- 
ent is  determined  by  the  balance  between  this  process 
and  the  process  of  generation,  whether  spontaneous  or 
due  to  radiation  ;  or  by  being  carried  out  by  the  proc- 

L  Mag.  (6),  4,  pp.  253-262,  1902. 


264  KINETIC   THEORY. 

ess  of  electrolytic  convection  and  discharge,  as  in  the 
experiments  on  the  conductivity  of  gases. 

In  some  experiments  as  in  the  "  electric  wind  "  from 
a  highly  charged  point 1  or  in  the  electrolysis  of  salt 
flames 2  there  is  room  for  question  as  to  the  mass  of 
the  carriers.  In  the  former  case  it  is  suggested  that 
the  corpuscle  may  carry  with  it  a  cluster  of  molecules 
electrically  attracted ;  in  the  latter  there  is  room  for 
question  whether  the  carriers  correspond  to  the  ions 
active  in  the  electrolysis  of  solutions  or  whether  they 
consist  of  negatively  charged  corpuscles  and  positively 
charged  residua,  molecules  which  have  lost  one  charged 
corpuscle. 

This  raises  the  general  question  whether  the  char- 
acter of  all  ions  may  not  be  due  to  an  excess  or  defi- 
ciency in  these  electrified  corpuscles,  anions  being 
characterized  by  an  excess  of  one  or  more  corpuscles, 
cations  by  the  corresponding  deficiency. 

1  Phil.  Mag.  (5),  48,  pp.  401-420,  1899. 
zPhil.  Mag.  (6),  4,  pp.  207-214,  1902. 


CHAPTER   XII. 
SUMMARY. 

THE  aim  of  this  work  as  announced  in  the  introduc- 
tion has  been  statement  rather  than  discussion.  In 
general  the  attempt  has  been  to  follow  along  lines 
which  are  tried  and  safe,  and  to  present  what  is  some- 
times termed  the  "orthodox"  treatment.  As  regards 
method  of  treatment,  a  middle  course  has  been  adopted  ; 
while  there  has  been  a  free  use  of  the  notation  and 
methods  of  the  differential  and  integral  calculus,  in  the 
attempt  to  give  a  presentation  which  should  be  suffi- 
ciently concrete  and  tangible  to  be  grasped  by  students 
in  college  such  abstractions  as  Clausius'  "  Virial 
theorem ' '  have  been  avoided  in  spite  of  their  great 
value  and  power.  On  the  other  hand,  in  the  frank 
recognition  of  the  ideal  character  of  the  system  under 
construction  numerical  details  with  regard  to  molecules 
and  their  phenomena  have  been  postponed  to  this 
chapter. 

Many  portions  of  the  theory  are  still  in  process  of 
construction,  many  points  are  still  debatable.  Max- 
well's distribution  of  velocities  has  both  its  defenders 
and  adversaries.  According  to  this  law  the  relative 
number  of  molecules  having  the  components  of  its 
velocity  u,  v,  and  w,  is  dependent  upon  the  function 

265 


266  KINETIC  THEORY. 


The  question  is  asked  whether  such  a  function  pos- 
sesses the  character  of  permanency.  Much  of  the 
work  of  such  masters  as  Burbury  and  Boltzmann  is  a 
discussion  of  this  fundamental  point,  the  latter  defend- 
ing the  law,  on  the  assumption  of  the  utter  lack  of 
systematic  relation  between  the  motions  of  the  different 
molecules  ("  molekular-ungeordnet "  is  his  phrase), 
while  the  former  insists  that  this  distribution  has  only 
a  quasi-stability,  complete  stability  being  given  by  a 
distribution  such  that  the  exponent  of  e  shall  be  a  com- 
plete quadratic  function,  involving  the  cross-products, 
uv,  vw>  wu,  with  suitable  coefficients.  Many  papers 
by  other  writers  also  are  taken  up  with  the  intricacies 
of  this  problem. 

Boltzmann's  defense  of  Maxwell's  distribution  of 
velocities  involves  his  famous  H  theorem.1  He  defines 
a  function  H,  such  that  for  a  single  gas,  if  the  number 
of  molecules  having  the  components  of  their  velocities 
between  u,  v,  w,  and  u  -f  du,  v  -}-  dv,  w  -f  dwy  be 
called  f  du  dv  dw 2 

H  =  ////  \ogfdu  dv  dw, 

and  the  criterion  of  the  stability  of  the  system  is  that 
//",  which  can  only  decrease  by  any  results  of  collisions, 
shall  be  a  minimum  and  hence  constant.  He  finds 

1  "Gastheorie,"  I.,  pp.  32-61. 

2  Notice  that  what  is  here  called  f  corresponds  to  the   expression 
nf(u}f{v}f(w}  in  Chapter  II.,  p.  20. 


SUMMARY.  267 

this  condition  satisfied  by  Maxwell's  distribution   for 
which 

n     -"2+"+™2         n     -*- 


and  hence 


so  that 


H=  log-yi  I    I    I  fdudvdw 2  I    I    I  fc^dudvdw 

but  the  value  of  the  first  integral  is  n,  and  of  the  sec- 
ond nc2,  while  c^ja?—  |,  so  that 


But  since  c2/a2  =  f  ,  J  Nmc*  =  RTandn  =  Njv, 

2~RT 


H 


f,          N 

=n(  log^i2 


=  n  log  (irlT-*)  +  Const. 

But  on  p.  52  we  have  shown  that  for  an  ideal  gas 
the  entropy  is 

5  =  Cv  log  T+  R  log  v  +  Const, 

Cy 

(22)  =  R  log  (vT*)  +  Const, 


268  KINETIC   THEORY. 

or  for  a  monatomic  gas,  for  which  CvjR  =  J, 
S=R\og(vT*)  +  Const. 

So  that  but  for  the  arbitrary  constant  term  the, 
entropy  appears  as  a  negative  multiple  of  //,  and  hence 
intimately  connected  with  the  stability  of  the  distribu- 
tion of  velocities,  and  with  the  impossibility  of  indi- 
vidual treatment  of  the  molecules.  In  this  connection 
it  is  worthy  of  note  that  the  entropy  was  defined  by 
the  equation 


in  which  i/7"was  an  integrating  factor,  so  that  5  was  a 
function  or  property  depending  only  on  the  state  of  the 
body,  and  the  equation 


rdQ 
L-T 


=  o, 


which  expresses  that  fact  is  also  the  mathematical  ex- 
pression of  the  second  law  of  thermo-dynamics,  which 
again  seems  to  depend  upon  our  inability  to  deal  indi- 
vidually with  molecules  ;  both  methods  of  discussion 
then  seem  to  point  to  a  relation  between  entropy  and 
the  character  of  the  molecular  motions. 

Another  bone  of  contention  is  the  doctrine  of  de- 
grees of  freedom,  stated  on  p.  75.  The  treatment 
there  given  yields  approximate  values  of  the  ratio  of  the 
two  specific  heats,  and  we  have  shown  (pp.  1  34—13  5)  that 
the  variation  between  the  behavior  of  actual  and  ideal 
gases  would  introduce  a  slight  corrective  factor,  the 


SUMMARY.  269 

computed  correction  being  of  the  same  order  as  the 
observed  variations.1  But  the  motions  there  consid- 
ered, of  translation  and  rotation,  are  not  the  only 
motions  conceivable  or  probable  to  a  molecule.  We 
can  add  relative  displacements  of  the  atoms,  and  atomic 
disturbances,  both  of  which  would  be  oscillatory,  peri- 
odic motions.  We  have  then  the  peculiar  fact  that  in 
counting  degrees  of  freedom  to  ascertain  the  distribu- 
tion of  energy,  in  investigating  specific  heats,  motions 
of  translation  and  rotation  are  to  be  considered,  but 
not  motions  of  vibration ;  that  is,  the  two  former 
classes  of  motions  are  so  intimately  related  that  in  the 
whole  body  of  gas  the  kinetic  energy  tends  to  dis- 
tribute itself  as  uniformly  among  all  their  degrees  of 
freedom  as  between  the  three  chosen  components  of 
the  translational  motion  ;  but  the  vibrational  motions 
seem  to  be  linked,  not  with  these  other  motions  but 
with  the  ether,  and  to  attain  their  equilibrium  mainly 
through  the  process  of  radiation.  Jeans 2  has  shown 
that  the  period  of  such  vibrations  is  so  small  in  com- 
parison with  the  probable  time  of  a  collision  that  the 
collisions  between  molecules  will  not  tend,  on  the 
whole,  to  produce  vibrations  sufficient  to  take  up  any 
considerable  proportion  of  the  energy ;  but  the  "  cor- 
puscles," with  their  very  much  lesser  size  and  higher 
speed  will  be  able  to  produce  such  vibrations.  This 
would  suggest  that  radiation  from  gases,  including 
luminosity,  is  largely  conditioned  upon  the  presence  of 

1  Phys.  Rev.,  XII.,  pp.  353-358,  1901. 

2  Phil.  Mag.  (6),  pp.  279-286,  1903. 


2/0  KINETIC   THEORY. 

considerable  numbers  of  these  corpuscles,  the  high 
temperature  increasing  the  vigor  of  their  attacks,  and 
the  readiness  of  their  formation. 

Another  question  of  interest  is  the  escape  of  gases 
from  our  atmosphere.  A  rough  computation  shows 
that  an  object  falling  from  an  infinite  distance  to  the 
surface  of  the  earth  ought  to  attain  a  speed  of  about 
one  million  cm.  per  sec.  We  found  (p.  14)  that  the 
average  speed  of  the  hydrogen  molecule  at  ordinary 
temperatures  was  a  little  less  than  two  hundred  thous- 
and cm.  per  sec.,  that  is,  a  little  less  than  a  fifth  of 
this  value.  At  the  "free  surface  of  the  atmosphere," 
if  we  were  to  regard  such  as  existing,  a  speed  not  much 
less  than  the  million  cm.  per  sec.  would  be  neces- 
sary for  the  escape  of  the  molecule  from  the  range  of 
the  earth's  attraction,  while  the  lower  temperature 
would  probably  lessen  the  actual  speeds  as  much  or 
more  relatively  than  the  greater  distance  from  the 
earth  would  lessen  the  speed  necessary  for  escape. 
The  figures  given  on  p.  31  would  indicate  that  for  so 
light  a  gas  as  hydrogen  less  than  one  molecule  in  io10 
would  have  sufficient  speed  to  escape,  while  the  heavier 
gases  of  the  atmosphere,  having  speeds  about  one 
fourth  as  great,  would  have  an  inconceivably  small 
chance  for  escape.  Any  considerable  loss  then  must 
be  confined  to  the  lighter  gases,  and  while  they  escape 
with  considerable  rapidity  from  the  immediate  neighbor- 
hood of  the  earth's  surface,  by  diffusion  and  convec- 
tion, it  is  problematical  whether  they  escape  absolutely 
from  the  earth's  atmosphere  to  any  great  extent. 


SUMMARY. 

In  the  development  of  the  theory  of  ideal  gases  it 
was  definitely  assumed  that  no  forces  acted  except 
during  that  portion  of  the  experience  of  a  molecule 
which  was  termed  a  collision,  and  no  assumption  was 
made  regarding  the  forces  there  developed  except  that 
the  restitutional  elasticity  was  perfect,  so  that  no  energy 
was  lost  in  the  collisions.  In  the  later  development  of 
van  der  Waals'  equation,  while  the  presence  of  attrac- 
tive forces  was  granted,  no  assumption  was  made,  ap- 
parently at  least,  further  than  that  the  forces  acting  in 
gases  were  of  the  same  nature  as  those  producing  sur- 
face tension  in  liquids. 

The  question  arises  whether  the  forces  resulting  in 
cohesion  and  capillarity  are  the  same  as  the  ordinary 
gravitational  attraction.  The  simplest  analytic  treat- 
ment of  capillarity  is  to  regard  the  liquid  as  a  homo- 
geneous medium,  compute  the  mutual  energy  of  two 
different  elements,  and  integrate  over  the  whole  vol- 
ume. Gauss,  van  der  Waals  and  others  have  found  for 
the  potential  energy  of  the  liquid  state  an  expression 
of  the  form  —  Ap  where  p  is  the  density  and  A  a  con- 
stant depending  upon  the  particular  liquid  and  the 
temperature.  Working  backwards  to  the  law  of  the 
force  between  the  individual  elements,  it  has  appeared 
not  to  be  of  the  simple  gravitational  type.  One  ex- 
planation offered  is  that  the  force  is  really  such  as  to 
be  expressed  by  a  mathematical  function  of  such  form 
that  for  finite  distances  it  has  the  common  Newtonian 
form.  Bakker l  has  suggested  for  the  potential  func- 

I Drude1  s  Annalen  (4),  n,  pp.  207-217,  1903. 


2/2  KINETIC   THEORY. 

tion  of  this  force,  instead  of  the  form  A/r,  the  more 
complicated 

A(r+<r)   -A_ 

_  /^r+<r 

r2 

in  which  X  is  a  distance  varying  inversely  as  the  tem  - 
perature,  of  such  an  order  that  at  finite  distances  the 
function  becomes  Newtonian.  Another  analytical  ex- 
pression for  the  force  would  be  a  series  of  the  form 


where  all  the  coefficients  after  the  first  are  so  small 
that  for  finite  distances  the  terms  drop  out,  giving  their 
effect  only  at  molecular  distances.  Many  have  sug- 
gested that  the  molecular  forces  vary  as  the  inverse 
fourth  power  of  the  distance.  This  relation  may  be 
deduced  in  rather  arbitrary  fashion  from  van  der  Waals' 
equation  as  follows  : 

Clausius'  virial  equation  may  be  written  * 


or  for  a  single  gas 

pv  = 

where  F  represents  the  force  acting  between  two  mole- 
cules, and  r  their  distance  apart,  the  summation  being 

i  van  der  Waals,  "  Continuitat  "  (2ded.),  I.,  p.  8,  eq.  (9). 


SUMMARY.  2/3 

taken  so  as  to  include  N  molecules.  Changing  the 
first  member  to  read  p(y  —  b),  and  comparing  with  van 
der  Waals'  equation, 


or  disregarding  b  as  small 


If  now 


But  the  values  of  r  are  evidently  proportional  to 
and  hence  v  is  proportional  to  r3,  and 


hence  the  correction  term  in  van  der  Waals'  equation 
would  ^eem  to  suggest  an  attraction  between  the  mole- 
cules, varying  inversely  as  the  fourth  power  of  the 
distance. 

Similarly  Boltzmann  !  has  developed  some  parts  of 
the  theoiy  on  the  assumption  that  the  phenomena  cf 

1<<Gastheorie,"  I.,  III.  Abschnitt,  pp.  153  204. 
18 


274  KINFTIC   THEORY. 

collision  are  due  to  a  repulsive  force  between  the 
molecules  proportional  to  the  fifth  power  of  the  dis- 
tance. 

But  to  state  the  mathematical  law  of  inter-molecular 
forces  only  tells  how  they  act,  not  why.  It  has  been 
suggested  by  many  thinkers  that  these  forces  may  be, 
in  part  at  least,  electrical.  This  suggestion  is  espe- 
cially pertinent  on  account  of  the  recent  vigorous  de- 
velopment of  the  theory  of  electrolytic  dissociation 
and  the  related  theory  of  electrons.  Thus  it  is  sug- 
gested that  the  electrical  forces  between  ions,  which 
have  electric  charges  of  one  sign,  are  proportional  to 
the  inverse  square  of  the  distance,  while  the  forces 
between  neutral  molecules,  which  are  supposed  to 
have  their  charges  not  neutralized  but  located  at  two 
near  points  within  the  molecule,  like  the  forces  be- 
tween other  electrical  and  magnetic  doublets,  must 
vary  as  the  inverse  fourth  power.  This  difference  in 
the  character  of  the  force  is  mentioned  l  as  a  possible 
explanation  of  discrepancies  between  results  of  con- 
ductivity determinations  and  freezing-  and  boiling-point 
methods  with  electrolytic  solutions. 

In  Chapter  IV.  we  have  developed  the  formula  for 
the  viscosity  of  a  gas 

(28)  77  ==  J  nmlc. 

The  coefficient  of  viscosity,  77,  can  be  determined  by 
experiment,  hence  we  can  find  the  mean  free  path,  /, 
which  is,  writing  for  nm  its  value  p,  the  density  of  the 
gas, 

LWetham,  Phil.  Mag.  (6),  5,  p.  285,  1903. 


SUMMARY.  275 


pc 

The  number  of  collisions  per  second  of  a  single  mole- 
cule is 

P=l/c. 

The  following  results  are  taken  from  O.  E.  Meyer  i1 


106 


TJ 

/ 

P 

Hydrogen, 

.000093 

.00001855  cm. 

9.48 

Nitrogen, 

184 

936     « 

4.76 

Oxygen, 

212 

1059     « 

4.07 

Carbon  monoxide, 

I84 

985     « 

4.78 

Carbon  dioxide, 

1  60 

680    " 

5-5i 

Chlorine, 

141 

474    " 

6.24 

Steam,  975  649    "  9.04      " 

Many  different  methods  have  been  employed  for 
finding  the  dimensions  of  the  molecules.  In  the  for- 
mula 


7T0-2  is  evidently  four  times  the  cross-sectional  area  of 
one  molecule,  and  mrcr2  that  of  all  the  molecules  in  a 
cubic  centimeter  of  gas,  so  the  ;/7rcr2/4  =  3/i6/  would 
be  approximately  the  area  covered  by  the  molecules 
if  arranged  in  close  order  in  a  single  layer.  For  the 
substances  in  the  preceding  table  these  areas  range  from 
9,500  for  hydrogen  to  37,300  for  chlorine,  that  is,  from 
one  to  four  square  meters,  approximately.  But  we 

1 "  Gastheorie  "    (edition  1877),  quoted  in  Winkelmann' s  "  Hand- 
buch,"  II.,  2,  p.  581. 


2/6  KINETIC   THEORY. 

have  shown  in  Chapter  VIII.  ,  p.  179,  that  the  minimum 
volume  which  could  be  attained  by  N  spherical  mole- 
cules was 


Applying  this  result  to  n  molecules  of  gas,  using  the 
other  value  of  the  mean  free  path 


(25) 

and  multiplying 


The  other  formula  for  /,  combined  with  the  value 
;/7r<73/6,  the  volume  of  the  n  spheres,  gives  the 
slightly  larger  value  often  quoted 

a-  =  S&J. 

Now  bv  the  least  possible  volume  occupied  by  what 
was  originally  one  cubic  centimeter  of  gas  under  atmos- 
pheric pressure,  as  a  result  of  the  greatest  pressure 
which  can  be  applied,  cannot  be  very  much  less  than 
the  volume  in  the  liquid  state.  On  the  assumption 
that  they  are  the  same,  Meyer  gives  the  following 
results,  computed  from  the  last  formula  : 

6l  <r 

Water,  .00081  44-10-8  Cm. 

Carbon  dioxide,  198  114  "          " 

Chlorine,  238  96  " 


SUMMARY.  2/7 

Again,  in  deducing   van  der  Waals'   equation,  we 
found  (p.  70), 


bl 
from  which  o-  =  2  — 

v 

which  gives  the  following  results  :  * 

b\v  <r 

Air,  .00387  56-10-9  cm. 

Nitrogen,  232  34     "  " 

Carbon  dioxide,  7  8  8     "  " 

Hydrogen,  318  88     "  " 

While  neither  of  these  methods  could  be  expected  to 
give  a  high  degree  of  accuracy,  and  the  first  particularly 
ought  to  give  results  too  large,  their  evidence  as  to  the 
order  of  magnitude  is  of  considerable  value.  Other  in- 
dependent methods  quoted  by  Jaeger  give  molecular 
diameters  of  the  same  order,  ranging  from  9  x  io~9  to 
70  x  io~9. 

Corroborative  evidence  is  furnished  by  measurements 
of  the  thickness  of  the  thinnest  films  which  are  able  to 
produce  certain  effects.  Thus  Quincke  found  that  a 
film  of  silver  of  the  thickness  5  x  io~6  cm.  affected  the 
the  adhesion  between  water  and  a  glass  plate.  Parks  2 
has  detected  films  of  water  condensed  on  the  surface  of 
glass,  ranging  from  7  x  io~6to  I3X  io~6  cm.  in  thick- 
ness. Johonnott3  has  measured  the  thickness  of  the 

1  Jaeger,  in  Winkelmann's  "  Handbuch  der  Physik,"  II.,  2,  p.  601. 
zPhil.  Mag.  (6),  5,  p.  518,  1903. 
•PAit.  Mag.  (5J,  47,  p.  522,  1899. 


278  KINETIC   THEORY. 

"black  spot"  in  a  soap-film,  supposed  to  be  twice  the 
range  of  molecular  attraction.  He  found  two  definite 
thicknesses,  that  of  the  "first  black  spot"  being 
n.2x  io~7  cm.,  of  the  "second  black  spot"  about  half 
as  great,  6.2  x  io~7.  These  results  are  of  considerable 
value  as  giving  a  large  upper  limit. 

Lord  Kelvin1  has  given  a  review  of  the  data  and 
some  valuable  conclusions.  He  says  :  "  It  is  scarcely 
conceivable  that  there  can  be  any  falling  off  in  the 
contractile  force"  of  a  water  film  "so  long  as  there 
are  several  molecules  in  the  thickness,"  and  that  con- 
sequently there  are  not  several  molecules  in  a  thick- 
ness of  io~8  cm.  He  quotes  the  work  of  Rayleigh 
and  Roentgen  on  thin  films  of  oil  on  the  surface  of 
clean  water.  The  former  found  the  motion  of  bits 
of  camphor  affected  by  a  film  10.6  x  io~8  cm.  thick, 
but  not  by  one  of  8.1  x  io~8  cm.  The  latter,  using 
ether,  was  able  to  detect  a  film  5.6  x  io~8  cm.  thick. 
Rayleigh  himself  suggests  that  these  thin  films  proba- 
bly contain  "  merely  molecules  of  oil  lying  at  greater 
and  less  distances  from  one  another,  but  at  no  part  of 
the  film  one  molecule  of  oil  lying  above  another  or 
resting  on  others." 

Kelvin2  gives  a  discussion  somewhat  similar  to  that 
just  given.  From  data  on  viscosity  he  finds  the  value 
of  nor2]  assuming  that  the  molecules  are  arranged  in 
the  liquid  state  in  cubic  order  with  distances  qcr  from 
center  to  center,  q  being  simply  a  ratio,  the  volume 

^Phil.  Mag.  (6),  4,  pp.  177  and  281,  1902. 
*Loc.  cit.,  p.  196. 


SUMMARY. 


279 


occupied  by  n  molecules  is  n(qcif.  Argon,  being 
monatomic,  seems  to  approach  more  nearly  the  ideal 
conditions  assumed  in  our  deductions,  of  hard  round 
molecules,  and  hence  seems  the  most  suitable  of  those 
gases  for  which  sufficient  data  are  available  for  testing 
the  theory.  He  gives  the  following  numerical  results  .' 

%<r)3=  1/681, 
no1  =  5/700, 

72=  68 12.  577ooV  =  8.9.  lo19?6, 

from  which  he  concludes,  since  q  is  likely  to  be  slightly 
greater  rather  than  less  than  unity,  to  give  reasonable 
mobility  to  the  liquid,  that  a  fair  value  for  n  is  io20. 
This  value  is  just  about  five  times  as  large  as  that  fre- 
quently quoted,  21  .  io18. 

On  the  assumption  of  this  value  of  n,  he  gives  the 
following  data : 


Gas. 

P 

c 

«<72 

<r 

m 

/ 

C02 

.001974 

39,200 

89,500 

2.99-ic-8 

19.74-10-24 

2-52-IO-6 

H2 
CO 

000090 
.001234 

184,200 
49,600 

32,900 
6l,3OO 

1.81 

2.48 

0.90 
12-34       ' 

6.84      " 
362      " 

N2 

.001257 

49,000 

6l,6oO 

248 

12-57 

3-64       " 

02 

001430 

46,100 

57.5°° 

2  40 

14.30 

3-91     " 

Argon 

.001781 

41,400 

57,700 

2.40 

17.81 

3.89    " 

It  will  be  seen  that  these  later  values  of  the  diameters 
of  molecules  are  of  the  same  order  as  those  found  by 
the  older  workers. 

J.  J.  Thomson,1  using  air  ionized  by  the  radiations 
from  radium,  counting  the  number  of  corpuscles  by 

*  Phil.  Mag.  (6),  5,  p.  354,  1903. 


280  KINETIC   THEORY. 

the  rate  of  fall  of  the  cloud  of  moisture  precipitated 
upon  them  by  a  suitable  expansion,  finds  the  mean 
value  of  the  ionic  charge  to  be  3.4-  io~10  electrostatic 
units.  H.  A.  Wilson  l  using  air  ionized  by  Roentgen 
rays  finds  the  very  similai  value  3.1  •  io~10.  Granting 
that  the  charge  on  an  ion  produced  by  radiation  is 
equal  to  that  on  the  hydrogen  ion  or  atom  in  solutions, 
as  shown  by  Townsend,2  these  results  give,  approxi- 
mately 

n  —  4-  io19 

about  two  fifths  of  the  value  of  n  as  found  by  Kelvin. 
In  the  last  chapter  the  work  of  Thomson  was  quoted 
showing  that  the  ratio  of  the  mass  to  the  charge  of  the 
gaseous  ion  was  about  io~r  (in  electromagnetic  units) 
while  for  the  hydrogen  ion  in  solution  it  is  about  icr*, 
so  that  the  mass  of  a  corpuscle  is  of  the  order  io~3  as 
compared  with  that  of  the  hydrogen  atom,  and  its  di- 
mensions of  the  order  of  i/io,  if  the  corpuscles  mak- 
ing up  the  atom  are  in  close  array.  Using  Kelvin's 
values  for  the  hydrogen  molecule  (two  atoms)  the  mass 
of  the  corpuscle  would  be  of  the  order  of  5  •  io~28,  a 
about  icr9,  c  about  8-io6  cm.  per  sec.,  and  the 
mean  free  path  in  hydrogen  at  o°  and  76  cm.  pressure 
about  icr6  cm.,  ?  thus  being  larger,  and  /  smaller  than 
for  the  molecules  of  the  gases. 


7.  Mag.  (6),  5,  p.  440,  1903. 

Phil.  Trans.,  A,  p.  129,  1899. 


INDEX. 


Absorption  of  gases,  189 

Acids,  strong,  241,  249 

Action,  spheres  of,  58 

Activity,  chemical,  related  to  ioni- 
zation,  249  ;  coefficient  of,  239, 
244 

Adiabatic,  47  ;  equation  of,  49 ; 
expansion  of  saturated  vapor, 
115  ;  for  substance  following 
van  der  Waals'  equation,  132 

Amagat,  experiments  on  high  pres- 
sures, 124 

Ammonium  chloride,  dissociation 
of,  242 

Andrews,  109 

Anion,  243 

Anode,  243 

Area  of  molecules,  275 

Argon,  279 

Arrangement  of  atoms,  76 

Arrhenius,  238  . 

Assumptions  of  elementary  theory,  7 

Atmosphere,  escape  of  gases  from, 
270 

Atoms,  3  ;  arrangement  of,  76 

Average  speed,  28 

Avogadro's  law,  43  ;  applied  to 
osmotic  pressure,  203 

l>,  value  of,  69,  70 
Bakker,  271 
Bases,  strong,  241,  249 
Bernouilli,  Daniel,  I 
Berthelot,  equations  of,  150 


Boiling  point,  elevation  of,  209 
Boltzmann,    I,   84,   89,    225,   266, 

273  ;    theorem   of   "degrees   of 

freedom,"  75;    H  theorem,  266 
Boyle's   law,    15  ;  variation   from, 

124  ;  osmotic  pressure  follows, 

202 
Burbury,  266 

Cailletet  and  Mathias,  rule  of,  143 

Carbon  dioxide,  isothermals  of,  109 

Carnot's  cycle,  47,  113 

Cathode,  243 ;  rays,  258,  260 ; 
rays,  deviation  of,  260 

Cation,  243 

Cell,  osmotic,  199 ;  galvanic, 
theory  of,  250  ;  irreversible  gal- 
vanic, 255 

Change  of  state,   thermodynamics, 

112 

Charge,  effect  of  electric,  on  con- 
densation of  moisture,  261  ;  on 
ions,  242,  261,  280;  on  mole- 
cules, 80 

Charles'  law,  15  ;  variation  from, 
124  ;  osmotic  pressure  follows, 
202 

Chemical  activity  related  to  ioniza- 
tion,  249 

Clausius,  I,  239,  265  ;  equation  of, 
123 

Cloud,  produced  in  ionized  air,  261 

Coefficient  of  activity,  239,  244  ;  of 
pressure-change,  126  ;  of  volume- 
Si 


282 


INDEX. 


change,  127  ;  of  viscosity,  87,  89; 
dependent  upon  size  of  mole- 
cules, 90;  upon  temperature,  91 ; 
variations  in,  92 

Cohesive  forces  in  fluid,  121 

Collisions,  8  ;  effect  on  distribution 
of  energy,  39  ;  of  single  molecule, 
56 ;  number  of,  59,  275  ;  in 
mixed  gas,  100  ;  number  causing 
dissociation,  227  ;  producing  re- 
combination, 229  ;  ionization  by, 
262 

Colored  ions,  240 

Component  velocities,  18,  33 

Compressibility  of  liquid,  184 

Concentration,  change  of,  in  elec- 
trolyte, 243  ;  -cells,  252 

Condensation  of  moisture  on  cor- 
puscles, 261 

Conduction  of  electricity,  79 ;  of 
heat,  92 

Conductivity,  electrical,  84  ;  ther- 
mal, 94  ;  dependent  on  tempera- 
ture, 95  ;  correction  at  surface, 
95  ;  of  electrolytes,  241,  244; 
molecular,  244,  246  ;  dependent 
on  viscosity,  248  ;  of  gas,  259 

Constant  pressure,  specific  heat  at, 

45 
Constant  volume,  specific  heat  at, 

44 

Constitution  of  water,  237 

Continuity  of  liquid  and  vapor 
states,  117 

Coordinates,  division  of  energy  be- 
tween, 37 

Copper,  solution  pressure  of,  255 

Corpuscles,  259  ;  production  of, 
262  ;  mass  of,  280  ;  mean  free 
path  of,  280 


Correction  of  thermal  conductivity, 

95 

Corresponding  states,  140 
Covolume,    152,    165  ;    of  liquid, 

181  ;  in  solution  and  surface  film, 

220 
Critical  data,  143  ;  point,  HO  ;  for 

van  der  Waals'   equation,    138  ; 

volume,  ideal,  145 
Crookes'  tube,  258 
Current,  electric,  83 
Curve  of  probabilities,  23 
Cycle,  Carnot's,  47,  113 

Dal  ton's  law,  39 
Daniell  cell,  253 
Data  of  critical  state,  143 
Decomposition,  double,  224 
Degrees  of  freedom,  75,  268 
Demon  engine,  Maxwell's,  54 
Density,  related  to  pressure,   13  ; 
relative,    cf    vapor   and    liquid, 
1 60,  165  ;  effect  upon  dissocia- 
tion,   231  ;    of  dissociated   gas, 
236  ;  effect   of,  on  electric  dis- 
charge in  gases,  263 
Depression  of  vapor  pressure,  207; 

of  freezing  point,  2 1 1 
Deviation  of  cathode  rays,  260 
Diameter  of  molecules,  276 
Dielectric  constant,  related  to  ion- 
izing power,  256 

Dieterici,   equation  of,    123,    147, 
149;  deduced,   171;  related   to 
equation  of  van  der  Waals,  1 73  ; 
treatment  of  vaporization,  154 
Diffusion  of  gases,    96;   "into  it- 
self," 99  ;  rate  of,  103';  simpli- 
fications, 105,  107 
Di  hydrol,  237 


INDEX. 


283 


Dilution,  heat  of,  213,  222 

Discharge,  electric,  in  gases,  258 

Dissociated  gas,  equations  for,  232  ; 
density  of,  236 

Dissociating  power  of  solvents,  256 

Dissociation,  gaseous,  224  ;  by  col- 
lision, 226  ;  into  like  parts,  227  ; 
collisions  causing,  227  ;  affected 
by  density,  231  ;  temperature  of, 
236  ;  electrolytic,  238  ;  constant, 
247  ;  of  water,  257 

Distance  travelled  by  molecule, 
64 ;  between  molecules  in  liquid, 
J75>  I78>  J8o,  278 

Distillation,  198 

Distribution  of  velocities,  18,  266; 
of  speeds,  25,  154,  164;  of 
energy  after  collision,  39 

Divisibility  of  matter,  2 

Division  of  energy  among  coordi- 
nates, 37 

Double  decomposition,  224 

Doublets,  274 

Efficiency  of  cycle,  114 
Electrical  conductivity,  84,  241 
Electric  charge  on  ion,  242  ;  effect 
on  condensation  of  moisture,  261 ; 
current,  83  ;  discharge  in  gases, 
258,    262  ;    spark,    262  ;  wind, 
264 

Electrical  forces,  274 
Electricity,  conduction  of,  79 
Electro-chemical  equivalent,  260 
Electrodes,  243 

Electrolyte,  change  of  concentra- 
tion, 243  ;  conductivity  of,  241, 
244 

Electrolytes,  specific  gravities  of, 
240 


Electrolytic  dissociation,  238 
Elevation  of  boiling  point,  209 
E.M.F.  of  galvanic  cell,  251,  254 
Energy,     division    among    coordi- 
nates, 37  ;  distribution  after  col- 
lision, 39  ;    intrinsic,  44  ;  inde- 
pendent   of    volume,     44 ;     of 
translation,   73  ;  total,   73  ;  car- 
ried by  molecules   passing   into 
vapor,  158  ;  potential,  of  liquid 
film,  170;  internal,  226;  poten- 
tial, of  liquid,  271 
Engine,    Carnot's  reversible,    47 ; 

Maxwell's  demon,  54 
Entropy  of  ideal  gas,  51,  268;  of 
saturated  vapor,  116;  of  sub- 
stance following  van  der  Waals' 
equation,  131  ;  of  mixed  gas, 
186 

Equation  of  adiabatic,  49  ;  of  van 
der  Waals,  122  ;  of  Clausius, 
123  ;  of  Dieterici,  123,  147, 
149,  171 

Equivalent,  electro-chemical,  260 
Escape  of  gases  from  atmosphere, 

270 
Exhaustion  of  conductivity  of  gas, 

259 
Expansion  of  saturated  vapor,  115 

Faraday's  law,  242 

Fifth  power,  inverse,  273 

Film,  semi -permeable,  200  ;  poten- 
tial energy  of  liquid,  170  ;  thick- 
ness of,  277 

First  law  of  thermodynamics,  43, 
213;  for  saturated  vapor,  114; 
for  substance  following  van  «1«r 
Waals'  equation,  129 

Flames,  conduction  in,  259 


284 


INDEX. 


Forces  between  molecules,  271 
Fourth  power,  inverse,  272 
Fractional  distillation,  198 
Freedom,  degrees  of,  75,  268 
Free  path,  mean,  55,  59,  64,  275  ; 
in  liquid,   176,  180  ;  of  corpus- 
cles, 262,  280 
Free  surface,  5 
Freezing  point,  depression  of,  211 

Galvanic   cell,    250 ;    irreversible, 

255 

Gas,  thermodynamics  of  ideal,  43  ; 
equations  for  ideal,  132  ;  for  one 
following  van  der  Waals'  equa- 
tion, 133  ;  equations  for  disso- 
ciated, 232  ;  density  of  disso- 
ciated, 236 ;  conductivity  of, 
259  ;  methods  of  ionizing,  263 

Gases,  ideal,  7  et  seq. ;  viscosity  of, 
85  ;  diffusion  of,  96  ;  do  not  fol- 
low laws  of  Boyle  and  Charles, 
1 20  ;  mixed,  185  ;  absorption  of, 
189  ;  ionization  of,  257  >  radia- 
tion from,  269 

Gaseous  dissociation,  224 ;  spec- 
trum, 259 

Gauss,  271 

Gay  Lussac's  law,  15 

Geissler  tube,  258 

Gravities,  specific,  of  electrolytes, 
240 

H  theorem,  266 

Half-electrolytes,  247 

Heat,  specific,  at  constant  volume, 
44  ;  at  constant  pressure,  45  ;  of 
molecule,  46,  71  ;  of  saturated 
vapor,  113  ;  at  constant  pressure, 
130  ;  of  isothermal  transforma- 


tion, 5°  5  conduction  of,  92  ; 
latent,  113,  169,  208,  210,  219; 
of  dilution,  213,  222  ;  specific 
and  latent,  of  water,  238 

Heats,  ratio  of  specific,  45  ;  value 
of  ratio,  77  ;  ratio  of  specific,  for 
substance  following  van  der 
Waals'  equation,  134  ;  ratio  of, 
268  ;  of  neutralization,  242 

Helmholtz,  250 

Henry's  law,  189 

Hydrodynamica,  Bernouilli's,  I 

Hydrogen,  behavior  of,  69  ;  escape 
of,  from  atmosphere,  270 

Hydrcl,  237 

Ice,  constitution  of,  237 

Ideal  gas,  7  »  pressure  of,  9 ; 
thermodynamics  of,  43  ;  equa- 
tions for,  132 

Ideal  isothermal,  116 

Impact,  momentum  transferred 
during,  10,  33  ;  at  right  angles, 
1 6  ;  probability  of,  56 

Impacts,  number  per  second,  1 1,  33 

Impulse,  summation  of,  35 

Indicators,  240 

Integration,  methods  of,  22,  27,  30 

Internal  pressure,  122,  153,  182 

Internal  energy,  226 

Inverse  fourth  power,  272  ;  fifth 
power,  273 

Ionic  charge,  242,  261,  280 

Ionization,  constant,  239,  244  ;  of 
water,  257  >  of  gases,  257  >  °f 
gas,  methods  of,  263 

Ionizing  power  of  solvents,  256 

Ions,  239  ;  their  properties  addi- 
tive, 239  ;  colored,  240  ;  valence 
of,  243  ;  migration  of,  244  ; 


INDEX. 


285 


velocity  of,  245  ;  speed  in  air, 
260  ;  mass  of,  280 

Isothermal,  47  ;  transformation, 
heat  and  work  of,  50  ;  ideal, 
116 

Isothermals  of  carbon  dioxide,  109; 
form  for  van  der  Waals'  equa- 
tion, 135 

Irreversible  galvanic  cell,  255 

Iso-osmotic  solutions,  202 

Jaeger,  277 
Jahn,  251 
Jeans,  269 
Johonnott,  277 
Joule,  I 

Kelvin,  72,  278 

Kinetic  theory  of  solutions,   216  ; 

of  dissociation,  225 
Kohlrausch,  245 

Latent  heat,  113,  169,  208,  210, 
219 

Latent  heats  of  water,  238 

Layer,  non-homogeneous,  155,  216 

Length  of  path,  probability  of,  64 

Limited  solubility,  192 

Liquid  state,  4  ;  film,  potential 
energy  of,  170;  molecules  within, 
174;  solutions,  189,  192;  po- 
tential energy  of,  271 

Lodge,  245 

Luminosity  of  gases,  269 

Magnesium,   solution  pressure   of, 

255 
Mass  of  ions,  264  ;  of  molecules, 

279  ;  of  corpuscles,  280 
Mariotte's  law,  15 


Maxwell,  I,  42 ;  distribution  of 
velocities,  21,  265  ;  distribution 
of  speeds,  25,  154,  164;  demon 
engine,  54  ;  deduction  of  relative 
speeds,  60  et  seq. 

"Mean  square"  of  speed,  29 

Mean  free  path,  55,  59,  64,  275  ; 
in  mixed  gas,  100,  103  ;  in 
liquid,  176,  1 80  ;  of  corpuscle, 
280 

Membrane,  semi -permeable,  200 

Method  of  integration,  22,  27,  30 

Meyer,  O.  E.,  107,  275 

Migration  of  ions,  244 

Milner,  169 

Minimum  volume,  179,  276 

Mixed  gas,  mean  free  path  in,  100, 
103 

Mixed  vapors,  194 

Mixture  of  gases,  pressure  of,  38 

Mixtures,  185 

Model,  ill 

Molecular  specific  heat,  46  ;  pres- 
sure, 122,  153  ;  conductivity, 
244,  246 

Molecule,  mean  kinetic  energy  pro- 
portional to  temperature,  42 

Molecules,  3  ;  oscillation  of,  4 ; 
speed  of,  14  ;  number  having 
different  speeds,  30;  number 
making  given  angle  with  a  plane, 
34  ;  effect  of  their  volume,  69  ; 
their  potentials  and  charges,  80  ; 
number  passing  into  vapor,  156  ; 
volume  of,  179  :  forces  between, 
271  ;  electrical  forces  between, 
274  ;  area  of,  275  ;  diameter  of, 
276  ;  mass  of,  279  ;  number  of, 
279 

Momentum  transferred  during  im- 


286 


INDEX. 


pact,   10,  33  ;  carried  by  mole- 
cule into  vapor,  161 

Most  probable  speed,  26 

Motions,  vibratory,  269 

Nernst,  153,  204,  241,  251,  256 

Neuclei,  ions  as,  261 

Neutralization,  heats  of,  242 

Newton's  laws  applicable  to  mole- 
cules, 3 

Newtonian  potential,  271 

Non-homogeneous  layer,  155  ;  film, 
216 

Noyes,  249 

Number  of  impacts  per  second,  II, 
33  ;  of  molecules  having  differ- 
ent speeds,  30  ;  of  collisions,  59, 
275  ;  of  collisions  causing  dis- 
sociation, 227  ;  of  collisions 
causing  recombination,  229  ;  of 
molecules  passing  into  vapor, 
156;  of  molecules,  279 

Ohm's  law  not  followed   by  gas, 

259 

Osmosis,  199 

Osmotic  pressure,  199,  222  ;  effect 
of  temperature,  201,  215  ;  re- 
lated to  vapor  pressure,  205  ; 
thermodynamics  of,  213  ;  anom- 
alous, 238 

Ostwald,  240,  247 

Parks,   277 

Partial  pressures  in  solution,  218 

Path,  mean  free,  55,  59,  64,  275  ; 

in   mixed    gas,     100,     103  ;    in 

liquid,  176,   1 80;  of  corpuscles, 

262,  280 
Path,  probability  of  given  length, 

64 


Pfeffer,  199 

Point,  critical,  no;  for  van  dei 
Waals'  equation,  138 

Polarization  of  galvanic  cell,  255 

Potential  of  molecules,  80  ;  energy 
of  liquid  film,  170 ;  energy  of 
liquid,  271 

Pressure  of  ideal  gas,  9,  13  ;  re- 
computed, 32  ;  due  to  several 
gases,  38  ;  modified  by  volume  of 
molecules,  67  ;  molecular,  122  ; 
internal,  153;  -change,  coefficient 
of,  126  ;  of  saturated  vapor,  by 
van  der  Waals'  equation,  137  ; 
critical,  139  ;  reduced,  140  ;  in 
liquid,  1 80  ;  of  vapor  over  solu- 
tion, 194  ;  osmotic  related  to 
vapor,  205  ;  osmotic,  222  ;  anom- 
alous osmotic,  238 ;  solution, 
252  ;  partial,  in  solution,  218 

Probabilities,  theory  of,  1 8 

Probability  curve,  23  ;  of  length  of 
path,  64 

Probable  speed,  26 

Quincke,  277 

Radiation  from  gases,  269 
Radiations     capable     of    ionizing 

gases,  263 

Ramsey  and  Young,  146 
Raoult's  law,  212,  223 
Ratio  of  two  specific  heats,  45,  77, 

268  ;  of  mass  to  charge,  260 
Rayleigh,  154,  278 
Rays,  cathode,  158,  260  ;  deviation 

of,  260 
Reactions,  chemical,  influenced  by 

presence  of  water,  249 
Recombination  of  molecules,  229 


INDEX. 


28; 


Reduced     piessure,    volume    and 

temperature,  140 
Relative  speed,  60  ;  in  mixed  gases, 

102 

Repulsive  forces,  273 

Reversible     transformations      and 

cycle,  47  ;  galvanic  cell,  250 
Roentgen,  278  ;  rays,  258 
Rudolphi,  248 
Rutherford,  258 

Saturated  vapor,    6  ;  specific  heat 
of,    113;  according   to   van  der 
Waals'  equation,  137 
Second  law  of  thermodynamics,  53? 

268 
Semi -permeable     partition,     186  ; 

film,  200 
Separation  of  mixed  gases,  1 86  ;  of 

solvent,  205 

Solid  state,  4  ;  solutions,  1 88 
Solubility,    249  ;  of  various  gases, 

190 

Solution  of  gases,  189  ;  of  liquids, 
192  ;  vapor  over,    193  ;   partial 
pressures  in,  218  ;    pressure,  252 
Solutions,  kinetic  theory  of,  216 
Solvent,  separation  of,  205 
Solvents,  ionizing  power  of,  256 
Space  occupied  by  molecules,  179 
Spark,  electric,  262 
Specific  gravities  of  electrolytes,  240 
Specific   heat  at  constant  volume, 
44  ;  at  constant  pressure,  45  ;  at 
constant  pressure  for   substance 
following  van  der  Waals'  equa- 
tion,  1 30  ;  of  molecule,  46  ;  of 
saturated  vapor,  113  ;  of  water, 
238 
Specific    heats,    ratio   of,  45,    77, 


268 ;  for  substance  following 
van  der  Waals'  equation,  134 

Spectrum,  gaseous,  259 

Speed  of  molecules,  14  ;  most  prob- 
able, 26;  average,  26;  "mean 
square, "  29  ;  relative,  60  ;  of 
ions,  243,  245  ;  of  cathode  rays, 
260 

Speeds,  distribution  of,  25,  154, 
164 ;  number  of  molecules 
having  different,  30  ;  relative,  in 
mixed  gas,  102  ;  in  liquid  and 
vapor,  1 60,  164 

Spheres  of  action,  58 

States,  corresponding,  140 

Statistical  method,  9 

Steam  line,  110 

Strong  acids  and  bases,  241,  249 

' '  Sugar  gas, ' '  204 

Sugar-solution,  osmotic  pressure  of, 

201 

Summation  of  impulses,  35 
Surface  film,    155,   216;  covolume 

in,  220 

Surface  tension,  183,  237 
Sutherland,     on      constitution     of 

water,  237 

Temperature,  scale  defined,  15  ; 
depends  on  mean  kinetic  energy 
of  molecules,  42  ;  critical,  139  ; 
reduced,  140 ;  of  dissociation, 
236 

Tension,  surface,  183,  237 
Thallous  chloride,  249 
Theory  of  Probabilities,  1 8 
Thermal  conductivity,  94  ;  depend- 
ent on  temperature,  95  ;  correc- 
tion at  surface,  95 
Thermodynamics,    scope,    2  ;    first 


288 


INDEX. 


law,  43  ;  of  ideal  gas,  43  ;  second 
law  of,  53,  268  ;  of  change  of 
state,  112;  of  substance  follow- 
ing van  der  Waals'  equation, 
128  ;  of  osmotic  pressure,  213  ; 
of  galvanic  cell,  250 

Thin  films,  277 

Thomson,  Prof.  James,  116 

Thomson,  J.  J.,  256,  258,  279 

Townsend,  280 

Total  energy,  73 

Transformation,  defined,  47  ;  work 
and  heat  of  isothermal,  50 

Translation,  energy  of,   73 

Traube,  152 

Tri-hydrol,  237 

Tube,  vacuum,  258 

Valence  of  ions,  243 

Valson,  240 

Values  of  ratio  of  specific  heats,  77 

Van  der  Waals,  27 1  ;  equation  of, 
122 ;  thermodynamics  of  sub- 
stance following  equation  of, 
128;  entropy,^/  131  ;  equations 
relating  to,  135  ;  ratio  of  speci- 
fic heats,  134/5  form  of  isother- 
mals,  135  ;  vabor  pressure,  137  ; 
critical  point,  \  138 

Van' t  Hoff,  202,  238,  248 

Vapor,  saturated,  6  ;  specific  heat 
of  saturated,  113;  adiabatic  ex- 
pansion of  saturated,  115  J  pres- 
sure according  to  van  der  Waals' 
equation,  137  ;  pressure,  re- 
duced, 141  ;  number  of  molecules 
passing  into,  156  ;  density,  160, 
165 ;  over  solution,  193 ;  pressure, 
osmotic  pressure  related  to,  205 

Vaporization,  5,  109,  155  ;  ther- 
modynamics of,  112 


Velocities,  distribution  of,  266 

Velocity  lines,  17;  number,  18 ; 
independent  of  direction,  20 ; 
Maxwell's  distribution,  21 

Velocity-function,  19 

Velocity  of  ions,  245 

Vibratory  motions,  72,  269 

Virial,  265,  272 

Viscosity  of  gases,  85,  274  ;  coeffi- 
cient of,  87,  89 

Volume,  of  molecules,  effect  of, 
69  ;  -change,  coefficient  of,  127  ; 
reduced,  140;  critical,  139,  145  ; 
in  liquid  state,  174;  minimum, 
179,  276  ;  of  molecules,  179 

Waals,  van  der,  271  ;  equation  of, 
122  ;  equations  for  substances 
following,  133  ;  ratio  of  specific 
heats,  134;  form  of  isothermals, 
!35  >  vapor  pressure,  137  ;  criti- 
cal point,  138 

Walker,  246 

Water  line,    no;  constitution   of, 

237  ;  specific  and   latent   heats, 

238  ;  undissociated,  241  ;   pres- 
ence in  chemical  reactions,  249  ; 
dissociation  of,  257 

Wetham,  204,  246,  257,  274 
Wilson,  C.  T.  R.,  261 
Wilson,  H.  A.,  280 
Wind,  electric,  264 
Work    of  isothermal    transforma- 
tions, 50  ;  of  vaporization,  167 

X-rays,  258 
Young,  141,  146 

Zehnder  tube,  258 

Zinc,  solution  pressure  of,  255 


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